Fokker-Planck Operators and the Center of the Enveloping Algebra
Bismut, Jean Michel
The heat equation method in index theory gives an explicit local formula for the index of a Dirac operator. Its Lagrangian counterpart involves supersymmetric path integrals. Similar methods can be developed to give a geometric formula for semi simple orbital integrals associated with the Casimir operator of a reductive group, this computation being related to Selberg's trace formula. The analogue of the heat equation method is now a suitable deformation of the Laplacian by a family of Fokker-Planck operators. In joint work with Shu SHEN, we have also shown how to extend these formulas to orbital integrals involving general elements of the center of the enveloping algebra.
 So it's a great honor and pleasure for me to come here for Samson's birthday. I mean we have been friends for how many years? I don't even know. 28. 28, more than that, no? 29. Maybe, okay. So I thought that it was a whole life anyway. So we have discussed about many things. I have listened to him on physics many, many times without being sure of understanding what he was saying. But I mean I sadly remember that when he talks to me he says, you're the unusual Frenchman. And I must say, I won't tell you what he refers to, but I'm very proud of this. Okay, so I will talk about Focke-Planck operators in the center of the enveloping algebra. So basically I mean I will explain certain things which are from physics point of view rather down to us, but maybe with applications to questions in mathematics which are of some interest. So I will basically explain how classical objects like cluplations can be deformed naturally to another category of operators which are in some sense operators well known in statistical physics like Focke-Planck operators. And this deformation is legitimacy. I mean it's not, you know, it's not for my own pleasure, but it was to connect with things like Zellberg's trace formula by an interpolation process. So the interpolation process I refer to is interpolation between the Laplacian on one hand and a dynamical system like the Joujizic flow on the other hand. And Zellberg's trace formula is exactly a sort of formula in which on one hand you have spectral data like Laplacian's and on the other hand you have Joujizic's and we will see our certain understanding of this formula will lead to explicit evaluations of objects which were not known before but using instruments which are in some sense taken from physics. So let me first of all start with extremely mental question, I mean considerations. So first of all the Euler characteristic, so you take X to be a compact remanian manifold and you know that the Euler characteristic is a global invariant that the alternates some of the dimensions of the homology groups of X. Now if you take diffeomorphism of X so it acts naturally on the homology so there is an action, linear action and so you can find the left set number which is again a global invariant which generalizes the Euler characteristic and which is just the alternates some of the traces of the diffeomorphism group acting on the homology. Now there is a way to compute explicitly this left set formula which is the so called fixed point formula or in the case of the identity Chern-Garbzman theorem but which is always based on the fact, I mean can be based on the fact that you can actually to compute the homology you can just use the Durand complex which homology is just the homology of X and if you introduce a remanian metric you introduce a former adjoint of the Durand operator and dx which is the little dx plus little dx star to be the corresponding drug operator and it's query is a hard-to-laplation. So basically the starting point of this sort of computation is to say that you can re-express left set number which is a global simple in some sense invariant in terms of super traces of operators which are now calculated on an infinite dimensional object and that the super trace of G exponential minus S times the Laplacian and this formula is true for any S. So in some sense if you just look at the right hand side there are many eigenvalues the contribution of zero eigenvalues of nonzero eigenvalues disappear which explains why the right hand side does not depend on S so that by making S tend to zero you obtain I mean you can obtain at least that the way of proving the left sets formula in terms of integral on fixed point set XG of the ailer form of TXG so if the points are isolated you just get a sum of plus or minus one. So let me now move to the real subject of my talk so I will consider now the heat operator on the manifold on the remanian compact remanian manifold X and I introduce again the Laplacian delta X and the corresponding heat current so now G for T positive it's the heat kernel at time T, T will be fixed so this is an operator which acts on smooth functions on X with values in R and so I will ask for rhetorical questions. So the first question is is the trace of the heat kernel a knowledge characteristic so can I think of this trace exactly in the way I was thinking of the trace of the difform of this before can I view this as sort of topological invariant so is this a generalize characteristic so the second question I will ask is exactly as before when I started from the chronology and I replaced the chronology by the Durand complex can I replace the smooth functions by a much bigger space by a much bigger space R such that I would write exactly formula of the same time as before re-expressing the trace of G in terms of a super trace of G but acting on a much bigger space equipped with a generalized Dirac operator which is exactly the analog of the operator DX which I introduced before and such that in a certain way I will instead of getting let's say churn Gauss-Bornay or the fixed point formulas when making B tend to infinity I would obtain something like Zellberg's trace formula which is an extension of Poisson formula so basically the difference with what I did before is now original space which was the chronology in the first example is now the space of smooth functions. So let me just formalize the analogy again so in the case of left shat number I had L of G which is left shat number generalized Euler characteristic on the right hand side I had a local formula for it which is a fixed point formula or churn Gauss-Bornay which is local and to prove the formula we use an interpolation involving the Durand complex and basically the interpolation just tells you that left hand side is ultimately equal to the right hand side. So this is exactly what we want to do in some sense I start on the trace of the heat kernel which is a global which is a global object and now of course I know more or less what I want to get on the right hand side which is Zellberg's trace formula but I want to find something in the middle which will do for us exactly what the Durand complex was doing before. So we are forced in some sense to consider the space of smooth functions in a manifold as the chronology of something the question is to know what is it the chronology of. So before going on but let me just remind you what Zellberg's trace formula is and a concrete example so this was the original formula proved by Zellberg so if X is a remand surface of constant scalar curvature minus 2 we introduce the closed udizix and their lengths L gamma. So Zellberg's trace formula in this case is an extension of Poisson formula so it expresses it expresses on the left hand side the trace of the heat kernel in terms of a bunch of terms the first term is a complicated one with an integral of r and with a volume factor in front of it the volume of x or it's not sigma is x and then there are a number of terms which are indexed by the closed udizix and with complicated factors so you see exponential minus a length square the usual thing in the denominator you have two hyperbolic sign of length square over two and then you have this infinite sum so in some sense what I will show is exactly finding what is a proper analytic and geometric structure which will interpolate between the two sides. So let me now move to the context on which I will effectively develop this theory so this will be the context of the symmetric spaces so I introduce G to be a real reductive group so if you don't know what reductive groups are just think of SL2R if you wish and k a maximal compact subgroup which is the case of SL2Y that's one and x equal g mod k is the corresponding symmetric space so I also introduce the carton splitting of the Lie algebra of the group so so so typically the Lie algebra splits into two pieces p and k there is a natural quadratic form of the Lie algebra which is positive on p and negative on k that's an orthogonal splitting and so this splitting of Lie algebras descends to a bundle of Lie algebras on the symmetric space Tx plus n modeled on the splitting g equal p plus k so again the basic example is g equal SL2R k it's just a matrix orthogonal matrix 2 by 2 matrix x is just a complex upper half plane and Tx plus n there is this n is a bundle of dimensions 3 over x so the analysis that we shall do will be done on the product of the group by its Lie algebra and quotient by k so quotient by k k acts on the group on the right it acts on the Lie algebra on the left and there is a standard procedure to do the quotient so the the analysis will be done on this and this subject is itself a geometric object so it is at the same time a group theoretic object but that's also a geometric object that the total space x x plus or hex hat of Tx plus n over x equal g mod k so you have this if you like the upper half plane you add this three dimensional bundle and so in the case of SL2 you will do the analysis I mean differential operators will occur on a space of total dimension 2 plus 3 which is 5 yes how is it the total bundle is trivial right the total bundle is trivial yes I mean the fact the bundle is trivial doesn't mean that the theory I will I mean it doesn't imply at least directly that the theory I will develop is trivial because the no but I mean I want to make the point that there will be a non-trivial coupling between the base and the fiber okay so we will do two separate constructions first of all the group and on the other side on the other hand on the Lie algebra so for the moment we forget about the quotient and only explain the two constructions on one hand on the group and on the other side on the Lie algebra so on the group side Kazimir and Kostard so if you have a reductive group like an acompactly group you have I mean you have the the the extension of the of the of the Laplacian or minus the Laplacian I would say I make it with a minus sign which is the Kazimir operator so that's an invariant differential operator on the group that takes into account the fact that there is this killing form which is positive in one direction negative on the other direction so that's why cg equal minus sum of e star i e i it takes into account the positivity or negativity of the form of the killing form B on the on the Lie algebra so if you like that an extension of Laplacian but that's a good Laplacian the p direction and a bad Laplacian with the wrong side in the k direction so that's why there are many Kazimir this is quadratic this is a quadratic Kazimir yes yes I will talk after all about all the possible Kazimir yes no I mean that will be the end of the talk okay so I introduce also the Clifford algebra of G equipped with a with an invariant form minus B and this Clifford algebra acts on the exterior algebra of G star so basically costant introduce an operator which whose square essentially is the is is the Kazimir okay so to do this I just introduce now the enveloping algebra U of G that the algebra of left invariant differential operators on the group so if you take the Lie algebra defines invariant vector fields and you look at the algebra differential operators generated by these vector fields at U of G and so the Dirac operator is a first order differential operator on the group and it can be described as a section of the Clifford algebra tensed by the by U of G is actually a differential operator order one of order one and it has first piece which looks like a Dirac operator the sum of c hat of E I star E I that's in some sense the classical Dirac operator and there is a correction which is a cubic correction of lengths three in the Clifford algebra and which involves the fundamental three form three form anti-symmetric three form on the Lie algebra which is given by B of U bracket VW that's an anti-symmetric three form and so that's the way that costant constructed is Dirac operator and so the formula of costant I remember that that that I was told by you that it was already proved by physicists before but that the square of the Dirac operator of costant is essentially up to a constant given by minus a Kazimier so this costant operator I mean doesn't act on smooth functions while the Kazimier act on smooth real functions the Kazimier acts on smooth function on the group with values in the exterior algebra of G star because of the Clifford algebra while Cg itself can be made to act on the smooth functions so this is such an identity we will use partly from an analytic point of view that is effectively try to reveal the hidden exterior algebra of G star now let me explain what we shall do on the Lie algebra side I told you the analysis is done on two pieces the group on one hand the Lie algebra and then we will couple them so on the Lie algebra G first of all we will make a weak rotation on the Lie algebra okay the Lie algebra G as I told you is not an Euclidean vector space it's a Lorentzian like vector space so we make a weak rotation to transform it into an Euclidean vector space so we replace p plus k by p plus i k and I introduced the harmonic oscillator on this gi so as you know the harmonic oscillator is given by this formula here where y is the generic is a generic point in gi and we will in some sense also on this gi introduce a Dirac operator a Dirac operator which is a Witten operator I mean that's that's connected with the Witten's idea of the twisting of the Dirac operator on a Euclidean vector space so we twist this on this Euclidean vector space the Dirac operator by the Gaussian by the Gaussian function so we get two new operators the modified Dirac which is d plus y wedge and the other one which is its swamble joint and these two operators also act now on smooth swarms on the Lie on on g whisk coefficients in the exterior algebra and so this operator here is a Dirac like operator acting on gi the sum of these two and the corresponding Laplacian that was a formula originally obtained by Witten tells you that up to a factor this Laplacian is given by the harmonic oscillator which is sort of bosonic number operator plus the fremianic number operator acting on the Lie algebra so we're going to couple these two constructions on one hand the costant operator acting on the group and on the other hand the Witten like operator acting on the Lie algebra so I will consider to construct now a new Dirac operator db parameterized by b positive so this operator db will be a combination of the Dirac operators on the group and on the Lie algebra so what does this mean this means that these new operator db will act on smooth functions on the group times the Lie algebra whisk coefficients in the exterior algebra of this time so remember I told you that the costant operator who is acting on functions on the group whiz values in the exterior algebra the Witten operator acts on functions on the Lie algebra whiz values in the same Lie algebra up to complexification so we can effectively combine the two so let me just give the formula so you see forget about the red factor so the first term in these db that's a new Dirac operator acting on a big space the first term is a costant Dirac operator and the second one is just a scaled version of the Witten like Dirac operator and there is a third mysterious term which is in red which is non-linear which is quadratic and whiz I mean whiz presence plays an essential role because if it was not there I would not be there either and I mean it's introduced it has to be forced because we will do the quotient thing by K remember that ultimately we'll descend everything to the quotient by K so that's this operator db so db is a K invariant operator it's not yes these whiz now correspond to the points in both directions separately or sorry sorry excuse me these whiz correspond to some points in the yes yes yes yes yes yes absolutely yes yes so in the gi direction you have two components yp and yk and that's why you take you take the league brackets okay so so that's exactly this so the the introduction of the quadratic term I won't have time to explain much about this is related you have to introduce these because you will eventually quotient things by K which introduces non-linearities by itself so what is this object that I will call the hyper elliptical aplation I will explain later what is hyper elliptic so it is just the operator lb you which is one half of db square minus dh d hat k square so remember that up to the quadratic factor I won't mention db is just a sum of the Dirac constant operator on g plus the written operator on the l algebra you take the square of the sum and you subtract in some sense the square of the first term so you undo what you had done except that you have less to is when you have a plus b square that a square plus ab plus ba plus b square so you subtract a square and you ultimately quotient the construction by K because all these subjects are K invariant they're not gene variant they are K invariant so as I mentioned before the le algebra descends to a vector bundle tx plus n which is sort of canonically trivial but not trivial as in the cladian vector bundle so let me introduce again the space x hat that's a bundle over x over the symmetric space whose fibers are tx plus n and so our operator lbx I explained before it will act on smooth sections over x script of the pullback of the exterior algebra of t star x plus n star so in other words g star is now replaced by t star x plus n star so using the bagman isomorphism I could put things in a more symmetric way instead of viewing things as a geometric object acting on this bigger space you could say I don't make the space bigger I just keep the space x down here but the price to pay is that you'll act on an infinite dimensional vector bundle involving the symmetric algebra which appeared also in untau minas talk this morning so it's the same it's the same object except I just have one copied set of having an infinite number of copies so let me give you first of all formula for this operator lbx so this operator lbx is supposed to be a deformation of the original scalar laplacian okay so this operator lbx if you just look at the red pieces it's essentially a forker plank operator so in other words this operator lbx it's harmonic oscillator in the fibers of tx plus n which I view as an eclidian vector bundle after weak rotation there is this harmonic oscillator in red and there is this piece here which is what I call the judizic flaw well if there was only tx if there was only tx you have a remanded manifold x you have the you have the fiber tx and on this big object of dimension twice the original dimension you have the judizic flaw and that's what's represented by the nabla y tx so if you forget about the other geometric terms we just have essentially a forker plank operator which is a coupling of this harmonic oscillator in the fiber and the judizic flaw in some sense which is a vector field on the base so this operator is non-self adjoint I mean it is not self adjoint obviously not the judizic flaw is anti symmetric it is it is not elliptic elliptic means that there is I mean it just differentiates once in the base direction not twice so it doesn't it's not a laplacian in the usual sense but what I'm saying is that from an analytic part of you it's a very good operator it's almost as good as the laplacian it's why it's called hyperliptic and what I'm saying besides is that this operator deforms naturally the original laplacian so this actually you can prove in a rigorous mathematical way this operator lbx deforms in some sense the original casimir operator or laplacian if you like acting on the base x so that means that when b10 to 0 there is a collapsing of the space x script to the space x so let's try to understand a little bit this collapsing I mean what happens is that when b10 to 0 the harmonic oscillator gets scaled by a big factor and that in some sense when b10 to 0 you ignore everything of the eigen eigen spaces or eigen values of this operator except the zero eigen value that's exactly a collapsing phenomenon which is in some sense exactly the same sort of collapsing that you have when you have look at calusacline theory in which you have very small circle the circle gets smaller and smaller because I mean in some sense you just concentrate on the fundamental on the ground state of the circle you have exactly the same phenomenon here you concentrate on the ground state of the harmonic oscillator except that there is here a non trivial coupling between the fiber and the base via the judizic floor which ultimately explains I won't prove that that in the proper sense the operator lbx deforms actually when b10 to 0 there is a language there is a theory in which you can say this is transport focal plan operator deforms the original laplacian but it is not only that it deforms it because it still captures a lot of information on the original laplacian in particular on its spectrum as long as it will work with compact quotients so as b10 to infinity after rescaling you see that it is the judizic floor which dominates so when b10 to 0 deforms to the laplacian when b10 to infinity the judizic floor dominates and if you think of taking traces of heat kernels you can see why when b10 to infinity you will ultimately select closed judizics so let me apply these two locally symmetric spaces now so original symmetric space x is just rn as a smooth space is just rn so we need to build first of all a compact manifold from the space so basically what you do is that you take a discrete subgroup of the group g and gamma and I will assume that it is co-compact and torsion free so this is just a complicated language to say that gamma acts freely on the in a discrete way on x so you can form the compact quotient gamma on x that's a compact manifold exactly like s1 is a quotient of r by the action of z and so I will give my first identity the first identity says that the trace on the smooth functions on z forget about this e just take e equals r for the moment on smooth functions of the heat kernel is equal to the super trace of the hyperliptic heat cut so we have exactly an identity which is similar to what I wanted to have before to deform the evaluation of the heat kernel into something more complicated involving more coordinates and actually if we think of it this idea of resolving of replacing this of saying the space of smooth functions is the cormology of something is actually embedded in the construction I gave before through this written harmonic oscillator I won't give details on that but we have exactly what we want sorry what is small c oh c is just the b square I mean that the defect in in in in in in costant's formula that just a defect in costant's formula that the lengths of rho g that's a square excuse me question about this this theorem yes so isn't that like the in physics you would say that you have a super symmetric s l2 sigma model and this is just localization yes so this is exactly so we have I mean in some sense we have constructed a super symmetry in a theory which is not super symmetric so that was yeah that's early algebra the algebra is where the fermions are and this is just localization the standard I mean localization seems not no I mean you you just we will localize like to rock okay we'll take b we'll take b b b tend to be busy and then we will be in it and then you get your physics yes yes that's so this is what we will do but sorry wasn't that done in the physics literature long ago no I don't think no not for that no no because I mean the idea that focal plan cooperators deform the Laplacian you see what this contains in particular is that the spectrum of the Laplacian is contained in the spectrum of these focal plan cooperators okay these focal plan cooperators have some real eigenvalues they have non real eigenvalues so ultimately when taking the super trace the non real eigenvalues disappear and what's left is just a real eigenvalues so you have had more eigenvalues but no okay so let me now say that there is a super symmetric interpretation involving involving the complication the complexification of the group but I won't mention it yet so splitting the identity so each side I mean since you're on a compact on a copent quotient by a discrete group each side splits as an infinite sum indexed by the conjugacy classes and gamma so this I mean exactly like Poisson original formula says that you take kernel on s1 is just a sum over z you can exactly do the same thing and now the miracle is that this identity splits as an identity that is you can forget about the group gamma and you could say the identity remains true in some sense element of gamma by element of gamma it is not an identity of sums the identity remains true and that's an identity of objects which are called orbital integrals so let me briefly mention what semi-simple orbital integrals are orbital integrals are fundamental in in in the theory of representation automorphic forms and so on so let me just take the simplest case actually I take a semi-simple element gamma and I call bracket gamma its conjugacy class so basically an orbital integrals that's its name is that you integrate the heat kernel on x I mean you need to introduce some notation for that on the orbit on the adjoint orbit of gamma by the action of g on the element gamma and the integral is an integral of the quotient of the quotient by z of gamma so these are the objects which are the ingredients which appear when you express the trace of the compact quotient in terms of the group gamma in some sense you re-express this trace as objects which are the orbital integrals so let me give a geometric description brief geometric description of what the orbital integrals are so I will just introduce this picture so this this is a flat space which is supposed to represent the symmetric space x equals rn if you like so inside this rn or inside the symmetric space there is a sub manifold which is x of gamma which is also a symmetric space and it can be geometrically constructed by minimizing by minimizing the so-called displacement function d gamma of x which is d of x and gamma x where d has a remanian distance now the remanian distance on such symmetric spaces is convex in the sense that it is convex on judizek so the minimizing set is non-empty because gamma is semi-simple and it is convex it's x of gamma the x of gamma is actually a symmetric space for the centralizer of gamma so how can you understand what the orbital integral is I just describe it geometrically what you do is that you take the heat kernel you take the heat kernel you take a normal you take an x0 in x of gamma you take a normal and you take a generic point why you look at its image bar gamma gamma is an azometry it maps normal judizek to a normal judizek and you find that because of negative curvature the mutual distance of y and gamma y grows at least linearly when y tends to infinity and you use ultimately the fact that the heat kernel decays at long distances like a Gaussian on the symmetric space so that the above integral exists so in other words you take the heat kernel at y and gamma y in integrated and normal directions you ultimately get a Gaussian integral which converges that's the orbital integral so the point of the hyperliptic theory will be to push ultimately what we will do is push the integral to x of gamma to the closed judizeks so the second fundamental identity is that effectively instead of having an identity of traces you get an identity of the so-called orbital integrals which are now calculated using heat kernels over this symmetric space not over the compact coefficient this is true for any b so let me now take the limit as b tends to infinity so after rescaling this operator lb the original operator we had when b tends to infinity is dominated in a way by the judizek flaw except that there is this extremely large term quartic term which dominates everything which means that localization will take place on commuting pairs yn and ytx and so ultimately the computation of the orbital integral localizes near the manifold of judizeks this x of gamma associated with gamma as I explained before the point of this is to push the integral to x of gamma so let me try to explain what is the structure of the formula for these orbital integrals so I take a gamma to be a semi-simple element which means that it is in a certain way diagonalizable so I take its reduced form I write it as a product of a self-adjoint if you like a unitary element e to the a times k minus one where a is in p and k is in k so after conjugation you can always do that and these pairs are commuting so let me call z of gamma to be the centralize of gamma and with le algebra splits again as p and gamma plus k of gamma so z of gamma is still is still a semi-simple is still a reductive group sorry yes just on the previous slide yes so I think it's related to the previous awkward question when you take the p to infinity yes this corresponds to the quarter of the flows right yes the judizeks yes so essentially it constitutes localization but your result comes from the deformed l right yes essentially this is not this is different for your plan plus your so so the localization will come from this deformed focacplug because you're taking the trace yeah but this result is not from the like conventional focacplug this is already deformed one because your lb is defined as a constant squared minus constant yes but I told you the original result tells you that the original object you went to compute is equal to the deformed one okay this was the first thing the original object you went to compute is equal to the form one so if you localize a deformed one you get something which is equal to the original object okay so you started from the casimir you deform it to this focacplug like operator but still you preserve the original object which means if you were working on the compact ocean the exact reproducers in this limit the same the same contribution sorry how about that so essentially in this limit it reproduces original result yes yes so it means like your deformation and yes there is a deformation but you do not change anything okay and and and now of course the result I have is no longer spectral result okay it's no longer the point is of this thing is if you can analyze now things judizik if you like close judizik by close judizik instead of having to localize an infinite number of judizik at the same time can you make a comment about yes the relationship to the physics yes because there is this paper on localization that you do the same formula by kecky or cacchina yes yes so what's the relation to what you're doing I don't know the paper that you mentioned okay maybe we'll talk about this later okay so let me give the evaluation of the semisimple orbital integrals and this formula will extend the formula of zelberg I gave at the beginning so there is an explicit function j gamma which is a function on i k of gamma so let's fix the notation again you have gamma semisimple element you have its the algebra z of gamma which splits into p and k of gamma and so this function here is defined on i k of gamma the weak rotated i k of gamma such that the original object the original object which is a trace gamma okay the orbital integral associated with the elliptic heat kernel and now let's look at the formula the formula should be looked at as an a tier singer like of index theorem in which integration on the manifold is replaced by integration on i k of gamma you have the first term which is j of gamma which will be an analog of the Todd or a roof genius but now it's a function now I have a churn character like if I need not introduce a twisting bundle so if you like you can ignore it if you're interested in sections of vector bundles you have to introduce it and you have a Gaussian integral here and so really the mystery lies in this function j of gamma so I told you it's a function it's not a characteristic class it's a function and so this function j of gamma it looks intimidating but in some sense it's a super symmetric version of the a roof genius on this manifold x you had these two bundles tx plus n but actually from the point of view of this point of view you have to look at this as tx minus m so the contribution will be the ratio of the corresponding a roof gene genera or a roof functions so I give the function here I give the formula which is every time a ratio of functions evaluated on the p and the k part the p part always appears in numerator the k part in denominator and these functions take into account the fine splitting of the Lie algebra under the action under the adjoint action of the representation of k of gamma the detail for the moment is not important here so the center of the enveloping algebra so again let me remind you that u of g is the enveloping algebra that's a non commutative algebra and inside there is a center which is a commutative algebra so I will introduce s of g which is the algebra of polynomials on g star and i of g which are the invariant polynomials on g star so the group the group there is an action let's say of the of the group or of the Lie algebra on these polynomials and look at these invariants so the Duflo isomorphism tells you that these two algebras are properly isomorphic so the isomorphism is not revealed you cannot guess what it is just by looking at these two algebras that it tells you that there is a canonical isomorphism of these two algebras there's a center of the Lie algebra is indeed an algebra of polynomials on g star so in particular for instance when you want to look at what corresponds to the casimir cg in this algebra i of g that's minus b star plus b star of rho g rho g b star being the du of b that is it's a polynomial of degree two on g star so center and enveloping algebras so and I take l to be an element of the center z of g and I take its counterpart in the algebra of polynomials i of g so what I'm saying is that this element 2d minus one of e this polynomial restricts to an element of i of z of gamma so the restriction is not entirely is not entirely natural but there is a natural projection from the Lie algebra on z of gamma so that you can define in some sense the proper sense the restriction of this element so from a polynomial you get a polynomial on the Lie algebra of the center and again from this polynomial now you rebuild a differential operator you rebuild a differential operator on z of gamma with constant coefficients so in other words you take the element of the center you look at the corresponding polynomial you restrict it to z of gamma and you rebuild a differential operator acting on z of gamma with constant coefficients I would say even acting on the complexification of z of gamma sorry the center yes z of gamma is just a Lie algebra z of the centralizer of gamma yeah the Lie algebra okay so the formula for general orbital integrals now instead of writing it for the heat kernel I take any I take any kernel in some sense of the Kazimier so this is a formula that I proved with Shushan in 2019 which says that the orbital integral of l so in other words a new ingredient now is this l this element of the center which actually probably involves all the possible Kazimiers that you were mentioning times a kernel an arbitrary kernel of the of the Kazimier well I mean it's given by the action of a new kernel you take a new kernel which now acts on the Lie algebra on i times k of gamma okay so you need to introduce to to understand this this formula is not too complicated you you have the action on the right hand side of the differential operator l z of gamma which acts on z of gamma and you compose it with a kernel of the corresponding Laplacian on z of gamma and you make it act on this distribution j gamma times in the case we introduce a twisting bundle the corresponding character for the twisting bundle we evaluate this kernel at zero so in other words you make it act on this distribution you have a smooth kernel you make it act on the distribution we evaluate it at zero and the formula is that the original orbital integral is equal to this new evaluation of smooth kernels on the Lie algebra of z of gamma okay so for l equals one that's essentially the original formula so the principle of the proof we start from the formula for this l equals one so when gamma is regular that is when it's regular that is if you like when all its eigenvalues are distinct so it centralizes a carton subgroup the formula I gave before for the heat kernel simplifies dramatically and can be differentiated in gamma and then ultimately we use fundamental results of a rich chandra to evaluate the action of a differential operator on our original formula the formula without the L to obtain a formula for l for gamma regular and so if gamma is semi-sibler or bitery we use a previous result for gamma regular and limit results of a rich chandra to obtain the general formula so let me just remind you what the function j gamma is so therefore general semi-simple element gamma that sort of quotients of determinants evaluated in the p and k part of the Lie algebra of z of gamma. Now, it turns out that when gamma is regular, they are sort of cancellations, I would say supersymmetric cancellations between the contributions of p and the contributions of k, so that the formula simplifies thematically. So, more precisely, you can re-express the function of j of gamma in terms of the roots of root system, but essentially of the purely imaginary roots of the reductive group. So, the imaginary roots are the ones we just correspond to the action of k on the p and k part, so that means that there is exactly as before where the eigenvalues were disappearing, here is the p and k factors which cancel each other. And so, this allows us to treat in some sense to prove that this function in the proper sense is smooth as a function of gamma, as long as gamma is regular, that is, if it's eigenvalues, if you like, are distinct. So, I will complete this by reminding you something that you all know, that the Langevin equation that in 1908 Langevin introduced the Langevin equation mx double dot equal minus x dot plus u double u dot to reconcile Bröhnian motion and classical mechanics. So, Bröhnian motion for m equals zero and classical mechanics for m equals infinity. So, in this theory of the interpolating hyperliptical appellation, m equals b square plays a role of a mass. So, this, I polyp, interpolation plays exactly, in some sense, uses the tools, all the objects introduced by Langevin. At the time, of course, there is no group theoretic interpretation. So, here is the, is a paper by Langevin in 1908 where you find exactly the equation written either what it before and with an interpretation that you all know. Okay, I think I will just give the references here and happy birthday, Samsung. Thank you very much. Yes? Yeah, I always have that main question about the constants Dirac. Yes. So, like if you took some other Dirac on the group, what would go wrong or what would go right? I mean, you see, there is the key thing which I have not explained is that the Kazimir lies in the center. Okay, so in the equality of the trace is equal to the super trace, the fact that it lies in the center plays a fundamental role because basically you need to prove that, you know, to write the super trace of some super commutator vanishes and the fact that the bad term disappears is related to the fact that it is a Kazimir. So, there is only one operator which will do what you want. All the other ones will always contain something which makes them outside of the center. That's somewhat complicated whatever book them. Yes, yes, yes. So, that's why you need absolutely the costum. Okay, so that's, I did not explain this, but the Kazimir is in the center. I mean, and that plays a fundamental role in the fact that the deformation is, yes? Two questions. Could you give an example of something that, you know, businesses like myself can understand that you can do with these wonderful formulas? Yes. And then the application, that's question one, question two is if you think of G as being, for example, the group of gauge transformations, can you use this in the DRS sphere? Yes, okay, so let me try to answer to this. Okay, so first of all about the role of the judizic flaw. Why is your judizic flaw there? And then I will give an explanation, I think, which is if you think of the judizic flaw and you think you write in coordinates, why do you have a value over dx? If you take a symbol that you should make a Fourier transform with the x variable, what you get is square root of minus one y xi. If you exponentiate this, you see that this is Fourier transform. Okay, so the ultimate reason why the judizic flaw appears, it's because in some sense it does for us mathematicians what it does is a Fourier transform. Okay, so we are geometrically trying to implement the idea of the Fourier transform in a context of course. If you think of Focke-Planck operators, I think it's fair to say nobody thinks of Focke-Planck operator of doing Fourier transform of anything. Okay, it is a dynamical, statistical, physics system that you try to describe, but the way we use it is this idea of Fourier transform. The second thing is that in the case where the face is flat, that is where you work in the case of r or s1, everything can be can be worked out explicitly in some sense. It is not so simple even that case to understand mathematically what happens. So let me try to say it in words. So you have the question that you know that you can quantize the group of simplectomorphisms. We can quantize it as a quantization to operators. You pass from simplectomorphism to unitary operators. Okay, let me say it roughly. What happens in this theory is that what you're quantizing is not simplectomorphism real ones, you're quantizing complex ones. Okay, so the quantization of complex simplectomorphism introduces a face. I mean if you look at the corresponding operator which appears on the other side, it will no longer be unitary, but the worst is that it will be badly unbounded. Okay, okay. However, if the consequences of this theory are true, that would mean that two operators which look very different are ultimately the same because they are conjugate by something. Okay, this something does not exist because it doesn't act, but still the operators behave behave properly. That is, they still have the same spectrum. Okay, so if you were doing this forgetting about this, you would just remain in the realm of classical Schrodinger-like operators. If you just quantize these non-complex transformations, you pass to a completely different category of operators. Yeah, so I think that the best explanation I can give for the moment. Other questions? Cannot put in certain functions, representations. Yes, yes, yes, you can put representation and that's why you have this extra contribution of pairing of the a-roof function. I mean, with a churn character function, which in this case is just the character corresponding to this representation. Yes, so that's why it was there. Yes. Other questions? If not, thanks again. Okay, thank you.
 So it's a great honor and pleasure for me to come here for Samson's birthday.  I mean we have been friends for how many years?  I don't even know.  28.  28, more than that, no?  29.  Maybe, okay.  So I thought that it was a whole life anyway.  So we have discussed about many things.  I have listened to him on physics many, many times without being sure of understanding  what he was saying.  But I mean I sadly remember that when he talks to me he says, you're the unusual Frenchman.  And I must say, I won't tell you what he refers to, but I'm very proud of this.  Okay, so I will talk about Focke-Planck operators in the center of the enveloping algebra.  So basically I mean I will explain certain things which are from physics point of view  rather down to us, but maybe with applications to questions in mathematics which are of some  interest.  So I will basically explain how classical objects like cluplations can be deformed  naturally to another category of operators which are in some sense operators well known  in statistical physics like Focke-Planck operators.  And this deformation is legitimacy.  I mean it's not, you know, it's not for my own pleasure, but it was to connect with  things like Zellberg's trace formula by an interpolation process.  So the interpolation process I refer to is interpolation between the Laplacian on one  hand and a dynamical system like the Joujizic flow on the other hand.  And Zellberg's trace formula is exactly a sort of formula in which on one hand you have  spectral data like Laplacian's and on the other hand you have Joujizic's and we will  see our certain understanding of this formula will lead to explicit evaluations of objects  which were not known before but using instruments which are in some sense taken from physics.  So let me first of all start with extremely mental question, I mean considerations.  So first of all the Euler characteristic, so you take X to be a compact remanian manifold  and you know that the Euler characteristic is a global invariant that the alternates  some of the dimensions of the homology groups of X.  Now if you take diffeomorphism of X so it acts naturally on the homology so there is  an action, linear action and so you can find the left set number which is again a global  invariant which generalizes the Euler characteristic and which is just the alternates some of the  traces of the diffeomorphism group acting on the homology.  Now there is a way to compute explicitly this left set formula which is the so called fixed  point formula or in the case of the identity Chern-Garbzman theorem but which is always  based on the fact, I mean can be based on the fact that you can actually to compute  the homology you can just use the Durand complex which homology is just the homology of X and  if you introduce a remanian metric you introduce a former adjoint of the Durand operator and  dx which is the little dx plus little dx star to be the corresponding drug operator and it's  query is a hard-to-laplation.  So basically the starting point of this sort of computation is to say that you can re-express  left set number which is a global simple in some sense invariant in terms of super traces  of operators which are now calculated on an infinite dimensional object and that the super  trace of G exponential minus S times the Laplacian and this formula is true for any S. So in some  sense if you just look at the right hand side there are many eigenvalues the contribution  of zero eigenvalues of nonzero eigenvalues disappear which explains why the right hand  side does not depend on S so that by making S tend to zero you obtain I mean you can obtain  at least that the way of proving the left sets formula in terms of integral on fixed point  set XG of the ailer form of TXG so if the points are isolated you just get a sum of  plus or minus one.  So let me now move to the real subject of my talk so I will consider now the heat operator  on the manifold on the remanian compact remanian manifold X and I introduce again the Laplacian  delta X and the corresponding heat current so now G for T positive it's the heat kernel  at time T, T will be fixed so this is an operator which acts on smooth functions on X with values  in R and so I will ask for rhetorical questions.  So the first question is is the trace of the heat kernel a knowledge characteristic so  can I think of this trace exactly in the way I was thinking of the trace of the difform  of this before can I view this as sort of topological invariant so is this a generalize characteristic  so the second question I will ask is exactly as before when I started from the chronology  and I replaced the chronology by the Durand complex can I replace the smooth functions  by a much bigger space by a much bigger space R such that I would write exactly formula  of the same time as before re-expressing the trace of G in terms of a super trace of G  but acting on a much bigger space equipped with a generalized Dirac operator which is  exactly the analog of the operator DX which I introduced before and such that in a certain  way I will instead of getting let's say churn Gauss-Bornay or the fixed point formulas when  making B tend to infinity I would obtain something like Zellberg's trace formula which is an extension  of Poisson formula so basically the difference with what I did before is now original space  which was the chronology in the first example is now the space of smooth functions.  So let me just formalize the analogy again so in the case of left shat number I had L  of G which is left shat number generalized Euler characteristic on the right hand side  I had a local formula for it which is a fixed point formula or churn Gauss-Bornay which is  local and to prove the formula we use an interpolation involving the Durand complex and basically  the interpolation just tells you that left hand side is ultimately equal to the right hand side.  So this is exactly what we want to do in some sense I start on the trace of the heat kernel  which is a global which is a global object and now of course I know more or less what I want to  get on the right hand side which is Zellberg's trace formula but I want to find something in the  middle which will do for us exactly what the Durand complex was doing before. So we are forced  in some sense to consider the space of smooth functions in a manifold as the chronology of  something the question is to know what is it the chronology of. So before going on but let me just  remind you what Zellberg's trace formula is and a concrete example so this was the original  formula proved by Zellberg so if X is a remand surface of constant scalar curvature minus 2  we introduce the closed udizix and their lengths L gamma. So Zellberg's trace formula in this case  is an extension of Poisson formula so it expresses it expresses on the left hand side the trace of  the heat kernel in terms of a bunch of terms the first term is a complicated one with an integral  of r and with a volume factor in front of it the volume of x or it's not sigma is x and then there  are a number of terms which are indexed by the closed udizix and with complicated factors so you  see exponential minus a length square the usual thing in the denominator you have two hyperbolic  sign of length square over two and then you have this infinite sum so in some sense what I will  show is exactly finding what is a proper analytic and geometric structure which will interpolate  between the two sides. So let me now move to the context on which I will effectively develop this  theory so this will be the context of the symmetric spaces so I introduce G to be a real reductive  group so if you don't know what reductive groups are just think of SL2R if you wish and k a maximal  compact subgroup which is the case of SL2Y that's one and x equal g mod k is the corresponding  symmetric space so I also introduce the carton splitting of the Lie algebra of the group so  so so typically the Lie algebra splits into two pieces p and k there is a natural quadratic form  of the Lie algebra which is positive on p and negative on k that's an orthogonal splitting  and so this splitting of Lie algebras descends to a bundle of Lie algebras on the symmetric space  Tx plus n modeled on the splitting g equal p plus k so again the basic example is g equal SL2R  k it's just a matrix orthogonal matrix 2 by 2 matrix x is just a complex upper half plane  and Tx plus n there is this n is a bundle of dimensions 3 over x so the analysis that we  shall do will be done on the product of the group by its Lie algebra and quotient by k so quotient  by k k acts on the group on the right it acts on the Lie algebra on the left and there is a standard  procedure to do the quotient so the the analysis will be done on this and this subject is itself  a geometric object so it is at the same time a group theoretic object but that's also a geometric  object that the total space x x plus or hex hat of Tx plus n over x equal g mod k so you have this  if you like the upper half plane you add this three dimensional bundle and so in the case of SL2  you will do the analysis I mean differential operators will occur on a space of total dimension  2 plus 3 which is 5 yes how is it the total bundle is trivial right the total bundle is trivial yes  I mean the fact the bundle is trivial doesn't mean that the theory I will I mean it doesn't imply at  least directly that the theory I will develop is trivial because the no but I mean I want to make  the point that there will be a non-trivial coupling between the base and the fiber okay so  we will do two separate constructions first of all the group and on the other side on the other  hand on the Lie algebra so for the moment we forget about the quotient and only explain the two  constructions on one hand on the group and on the other side on the Lie algebra so on the group  side Kazimir and Kostard so if you have a reductive group like an acompactly group you have I mean you  have the the the extension of the of the of the Laplacian or minus the Laplacian I would say I make  it with a minus sign which is the Kazimir operator so that's an invariant differential operator on  the group that takes into account the fact that there is this killing form which is positive in  one direction negative on the other direction so that's why cg equal minus sum of e star i e i it  takes into account the positivity or negativity of the form of the killing form B on the on the Lie  algebra so if you like that an extension of Laplacian but that's a good Laplacian the p direction  and a bad Laplacian with the wrong side in the k direction so that's why there are many Kazimir  this is quadratic this is a quadratic Kazimir yes yes I will talk after all about all the possible  Kazimir yes no I mean that will be the end of the talk okay so I introduce also the Clifford  algebra of G equipped with a with an invariant form minus B and this Clifford algebra acts on the  exterior algebra of G star so basically costant introduce an operator which whose square essentially  is the is is the Kazimir okay so to do this I just introduce now the enveloping algebra U of G  that the algebra of left invariant differential operators on the group so if you take the Lie  algebra defines invariant vector fields and you look at the algebra differential operators  generated by these vector fields at U of G and so the Dirac operator is a first order differential  operator on the group and it can be described as a section of the Clifford algebra tensed by the  by U of G is actually a differential operator order one of order one and it has first piece  which looks like a Dirac operator the sum of c hat of E I star E I that's in some sense the  classical Dirac operator and there is a correction which is a cubic correction of lengths three in  the Clifford algebra and which involves the fundamental three form three form anti-symmetric  three form on the Lie algebra which is given by B of U bracket VW that's an anti-symmetric  three form and so that's the way that costant constructed is Dirac operator and so the formula  of costant I remember that that that I was told by you that it was already proved by physicists  before but that the square of the Dirac operator of costant is essentially up to a constant given  by minus a Kazimier so this costant operator I mean doesn't act on smooth functions while the  Kazimier act on smooth real functions the Kazimier acts on smooth function on the group  with values in the exterior algebra of G star because of the Clifford algebra while Cg itself  can be made to act on the smooth functions so this is such an identity we will use partly from  an analytic point of view that is effectively try to reveal the hidden exterior algebra of G star  now let me explain what we shall do on the Lie algebra side I told you the analysis is done on  two pieces the group on one hand the Lie algebra and then we will couple them so on the Lie algebra  G first of all we will make a weak rotation on the Lie algebra okay the Lie algebra G as I told you  is not an Euclidean vector space it's a Lorentzian like vector space so we make a weak rotation to  transform it into an Euclidean vector space so we replace p plus k by p plus i k and I introduced  the harmonic oscillator on this gi so as you know the harmonic oscillator is given by this formula  here where y is the generic is a generic point in gi and we will in some sense also on this gi  introduce a Dirac operator a Dirac operator which is a Witten operator I mean that's that's connected  with the Witten's idea of the twisting of the Dirac operator on a Euclidean vector space  so we twist this on this Euclidean vector space the Dirac operator by the Gaussian by the Gaussian  function so we get two new operators the modified Dirac which is d plus y wedge and the other one  which is its swamble joint and these two operators also act now on smooth swarms on the Lie on on  g whisk coefficients in the exterior algebra and so this operator here is a Dirac like operator  acting on gi the sum of these two and the corresponding Laplacian that was a formula  originally obtained by Witten tells you that up to a factor this Laplacian is given by the harmonic  oscillator which is sort of bosonic number operator plus the fremianic number operator acting on the  Lie algebra so we're going to couple these two constructions on one hand the costant operator  acting on the group and on the other hand the Witten like operator acting on the Lie algebra  so I will consider to construct now a new Dirac operator db parameterized by b positive  so this operator db will be a combination of the Dirac operators on the group and on the Lie  algebra so what does this mean this means that these new operator db will act on smooth functions  on the group times the Lie algebra whisk coefficients in the exterior algebra of this time so remember  I told you that the costant operator who is acting on functions on the group whiz values in the  exterior algebra the Witten operator acts on functions on the Lie algebra whiz values in the  same Lie algebra up to complexification so we can effectively combine the two so let me just give  the formula so you see forget about the red factor so the first term in these db that's a new Dirac  operator acting on a big space the first term is a costant Dirac operator and the second one is  just a scaled version of the Witten like Dirac operator and there is a third mysterious term  which is in red which is non-linear which is quadratic and whiz I mean whiz presence plays an  essential role because if it was not there I would not be there either and I mean it's introduced  it has to be forced because we will do the quotient thing by K remember that ultimately we'll descend  everything to the quotient by K so that's this operator db so db is a K invariant operator it's  not yes these whiz now correspond to the points in both directions separately or sorry sorry  excuse me these whiz correspond to some points in the yes yes yes yes yes yes absolutely yes yes  so in the gi direction you have two components yp and yk and that's why you take you take the  league brackets okay so so that's exactly this so the the introduction of the quadratic term I  won't have time to explain much about this is related you have to introduce these because you will  eventually quotient things by K which introduces non-linearities by itself so what is this object  that I will call the hyper elliptical aplation I will explain later what is hyper elliptic  so it is just the operator lb you which is one half of db square minus dh d hat k square so remember  that up to the quadratic factor I won't mention db is just a sum of the Dirac constant operator on g  plus the written operator on the l algebra you take the square of the sum and you subtract in  some sense the square of the first term so you undo what you had done except that you have less  to is when you have a plus b square that a square plus ab plus ba plus b square so you  subtract a square and you ultimately quotient the construction by K because all these subjects  are K invariant they're not gene variant they are K invariant so as I mentioned before the  le algebra descends to a vector bundle tx plus n which is sort of canonically trivial but not  trivial as in the cladian vector bundle so let me introduce again the space x hat that's a bundle  over x over the symmetric space whose fibers are tx plus n and so our operator lbx I explained before  it will act on smooth sections over x script of the pullback of the exterior algebra of t star x  plus n star so in other words g star is now replaced by t star x plus n star so using the  bagman isomorphism I could put things in a more symmetric way instead of viewing things as a  geometric object acting on this bigger space you could say I don't make the space bigger I just  keep the space x down here but the price to pay is that you'll act on an infinite dimensional  vector bundle involving the symmetric algebra which appeared also in untau minas talk this morning  so it's the same it's the same object except I just have one copied set of having an infinite number  of copies so let me give you first of all formula for this operator lbx so this operator lbx is  supposed to be a deformation of the original scalar laplacian okay so this operator lbx if you  just look at the red pieces it's essentially a forker plank operator so in other words this  operator lbx it's harmonic oscillator in the fibers of tx plus n which I view as an eclidian  vector bundle after weak rotation there is this harmonic oscillator in red and there is this  piece here which is what I call the judizic flaw well if there was only tx if there was only tx  you have a remanded manifold x you have the you have the fiber tx and on this big object of dimension  twice the original dimension you have the judizic flaw and that's what's represented by the nabla y tx  so if you forget about the other geometric terms we just have essentially a forker plank operator  which is a coupling of this harmonic oscillator in the fiber and the judizic flaw in some sense  which is a vector field on the base so this operator is non-self adjoint I mean it is not  self adjoint obviously not the judizic flaw is anti symmetric it is it is not elliptic elliptic means  that there is I mean it just differentiates once in the base direction not twice so it doesn't  it's not a laplacian in the usual sense but what I'm saying is that from an analytic part of you  it's a very good operator it's almost as good as the laplacian it's why it's called hyperliptic  and what I'm saying besides is that this operator deforms naturally the original laplacian  so this actually you can prove in a rigorous mathematical way this operator lbx deforms in  some sense the original casimir operator or laplacian if you like acting on the base x so that means  that when b10 to 0 there is a collapsing of the space x script to the space x so let's try to  understand a little bit this collapsing I mean what happens is that when b10 to 0 the harmonic  oscillator gets scaled by a big factor and that in some sense when b10 to 0 you ignore everything  of the eigen eigen spaces or eigen values of this operator except the zero eigen value that's exactly  a collapsing phenomenon which is in some sense exactly the same sort of collapsing that you have  when you have look at calusacline theory in which you have very small circle the circle gets smaller  and smaller because I mean in some sense you just concentrate on the fundamental on the ground state  of the circle you have exactly the same phenomenon here you concentrate on the ground state of the  harmonic oscillator except that there is here a non trivial coupling between the fiber and the base  via the judizic floor which ultimately explains I won't prove that that in the proper sense the  operator lbx deforms actually when b10 to 0 there is a language there is a theory in which you can  say this is transport focal plan operator deforms the original laplacian but it is not only that  it deforms it because it still captures a lot of information on the original laplacian in particular  on its spectrum as long as it will work with compact quotients so as b10 to infinity after  rescaling you see that it is the judizic floor which dominates so when b10 to 0 deforms to the  laplacian when b10 to infinity the judizic floor dominates and if you think of taking traces of  heat kernels you can see why when b10 to infinity you will ultimately select closed judizics so  let me apply these two locally symmetric spaces now so original symmetric space x is just rn as  a smooth space is just rn so we need to build first of all a compact manifold from the space so  basically what you do is that you take a discrete subgroup of the group g and gamma and I will assume  that it is co-compact and torsion free so this is just a complicated language to say that gamma acts  freely on the in a discrete way on x so you can form the compact quotient gamma on x that's a  compact manifold exactly like s1 is a quotient of r by the action of z and so I will give my first  identity the first identity says that the trace on the smooth functions on z forget about this e  just take e equals r for the moment on smooth functions of the heat kernel is equal to the  super trace of the hyperliptic heat cut so we have exactly an identity which is similar to what I  wanted to have before to deform the evaluation of the heat kernel into something more complicated  involving more coordinates and actually if we think of it this idea of resolving of replacing  this of saying the space of smooth functions is the cormology of something is actually embedded  in the construction I gave before through this written harmonic oscillator I won't give details  on that but we have exactly what we want sorry what is small c oh c is just the b square I mean  that the defect in in in in in in costant's formula that just a defect in costant's formula that the  lengths of rho g that's a square excuse me question about this this theorem yes so isn't that like the  in physics you would say that you have a super symmetric s l2 sigma model and this is just  localization yes so this is exactly so we have I mean in some sense we have constructed a super  symmetry in a theory which is not super symmetric so that was yeah that's early algebra the algebra  is where the fermions are and this is just localization the standard I mean localization  seems not no I mean you you just we will localize like to rock okay we'll take b we'll take b b b  tend to be busy and then we will be in it and then you get your physics yes yes that's so this is what  we will do but sorry wasn't that done in the physics literature long ago no I don't think  no not for that no no because I mean the idea that focal plan cooperators deform  the Laplacian you see what this contains in particular is that the spectrum of the Laplacian  is contained in the spectrum of these focal plan cooperators okay these focal plan cooperators  have some real eigenvalues they have non real eigenvalues so ultimately when taking the super  trace the non real eigenvalues disappear and what's left is just a real eigenvalues so you  have had more eigenvalues but no okay so let me now say that there is a super symmetric  interpretation involving involving the complication the complexification of the group but I won't  mention it yet so splitting the identity so each side I mean since you're on a compact  on a copent quotient by a discrete group each side splits as an infinite sum indexed by the  conjugacy classes and gamma so this I mean exactly like Poisson original formula says that you  take kernel on s1 is just a sum over z you can exactly do the same thing and now the miracle  is that this identity splits as an identity that is you can forget about the group gamma  and you could say the identity remains true in some sense element of gamma by element of gamma  it is not an identity of sums the identity remains true and that's an identity of objects which are  called orbital integrals so let me briefly mention what semi-simple orbital integrals are  orbital integrals are fundamental in in in the theory of representation automorphic forms and  so on so let me just take the simplest case actually I take a semi-simple element gamma  and I call bracket gamma its conjugacy class so basically an orbital integrals that's its name  is that you integrate the heat kernel on x I mean you need to introduce some notation for that  on the orbit on the adjoint orbit of gamma by the action of g on the element gamma and the  integral is an integral of the quotient of the quotient by z of gamma so these are the objects  which are the ingredients which appear when you express the trace of the compact quotient in terms  of the group gamma in some sense you re-express this trace as objects which are the orbital integrals  so let me give a geometric description brief geometric description of what the orbital  integrals are so I will just introduce this picture so this this is a flat space which is  supposed to represent the symmetric space x equals rn if you like so inside this rn or  inside the symmetric space there is a sub manifold which is x of gamma which is also a symmetric  space and it can be geometrically constructed by minimizing by minimizing the so-called  displacement function d gamma of x which is d of x and gamma x where d has a remanian distance  now the remanian distance on such symmetric spaces is convex in the sense that it is convex  on judizek so the minimizing set is non-empty because gamma is semi-simple and it is convex  it's x of gamma the x of gamma is actually a symmetric space for the centralizer of gamma  so how can you understand what the orbital integral is I just describe it geometrically  what you do is that you take the heat kernel you take the heat kernel you take a normal you take  an x0 in x of gamma you take a normal and you take a generic point why you look at its image  bar gamma gamma is an azometry it maps normal judizek to a normal judizek and you find that because  of negative curvature the mutual distance of y and gamma y grows at least linearly when y tends  to infinity and you use ultimately the fact that the heat kernel decays at long distances  like a Gaussian on the symmetric space so that the above integral exists so in other words you take  the heat kernel at y and gamma y in integrated and normal directions you ultimately get a  Gaussian integral which converges that's the orbital integral so the point of the hyperliptic  theory will be to push ultimately what we will do is push the integral to x of gamma to the closed  judizeks so the second fundamental identity is that effectively instead of having an identity of  traces you get an identity of the so-called orbital integrals which are now calculated  using heat kernels over this symmetric space not over the compact coefficient this is true for any b  so let me now take the limit as b tends to infinity so after rescaling this operator lb the original  operator we had when b tends to infinity is dominated in a way by the judizek flaw except  that there is this extremely large term quartic term which dominates everything which means that  localization will take place on commuting pairs yn and ytx and so ultimately the computation of the  orbital integral localizes near the manifold of judizeks this x of gamma associated with gamma as  I explained before the point of this is to push the integral to x of gamma so let me try to explain  what is the structure of the formula for these orbital integrals so I take a gamma to be a  semi-simple element which means that it is in a certain way diagonalizable so I take its reduced  form I write it as a product of a self-adjoint if you like a unitary element e to the a times k  minus one where a is in p and k is in k so after conjugation you can always do that and these pairs  are commuting so let me call z of gamma to be the centralize of gamma and with le algebra  splits again as p and gamma plus k of gamma so z of gamma is still is still a semi-simple is still  a reductive group sorry yes just on the previous slide yes so I think it's related to the previous  awkward question when you take the p to infinity yes this corresponds to the quarter of the flows  right yes the judizeks yes so essentially it constitutes localization but your result comes  from the deformed l right yes essentially this is not this is different for your plan plus your  so so the localization will come from this deformed focacplug because you're taking the trace  yeah but this result is not from the like conventional focacplug this is already deformed one  because your lb is defined as a constant squared minus constant yes but I told you the original  result tells you that the original object you went to compute is equal to the deformed one okay this  was the first thing the original object you went to compute is equal to the form one so if you  localize a deformed one you get something which is equal to the original object okay so you started  from the casimir you deform it to this focacplug like operator but still you preserve the original  object which means if you were working on the compact ocean the exact reproducers in this limit the same  the same contribution sorry how about that so essentially in this limit it reproduces  original result yes yes so it means like your deformation and yes there is a deformation  but you do not change anything okay and and and now of course the result I have is no longer  spectral result okay it's no longer the point is of this thing is if you can analyze now things  judizik if you like close judizik by close judizik instead of having to localize an infinite number  of judizik at the same time can you make a comment about yes the relationship to the physics  yes because there is this paper on localization that you do the same formula by  kecky or cacchina yes yes so what's the relation to what you're doing  I don't know the paper that you mentioned okay maybe we'll talk about this later okay so let me give  the evaluation of the semisimple orbital integrals and this formula will extend the formula of  zelberg I gave at the beginning so there is an explicit function j gamma which is a function on  i k of gamma so let's fix the notation again you have gamma semisimple element you have its  the algebra z of gamma which splits into p and k of gamma and so this function here is defined on  i k of gamma the weak rotated i k of gamma such that the original object the original object which  is a trace gamma okay the orbital integral associated with the elliptic heat kernel  and now let's look at the formula the formula should be looked at as an a tier singer like of index  theorem in which integration on the manifold is replaced by integration on i k of gamma  you have the first term which is j of gamma which will be an analog of the Todd or a roof genius  but now it's a function now I have a churn character like if I need not introduce a twisting bundle so  if you like you can ignore it if you're interested in sections of vector bundles you have to introduce  it and you have a Gaussian integral here and so really the mystery lies in this function  j of gamma so I told you it's a function it's not a characteristic class it's a function  and so this function j of gamma it looks intimidating but in some sense it's a super  symmetric version of the a roof genius on this manifold x you had these two bundles tx plus n  but actually from the point of view of this point of view you have to look at this as tx minus m  so the contribution will be the ratio of the corresponding a roof gene genera or a roof  functions so I give the function here I give the formula which is every time a ratio of functions  evaluated on the p and the k part the p part always appears in numerator the k part in denominator  and these functions take into account the fine splitting of the Lie algebra under the action  under the adjoint action of the representation of k of gamma the detail for the moment is not important  here so the center of the enveloping algebra so again let me remind you that u of g is the  enveloping algebra that's a non commutative algebra and inside there is a center which is a commutative  algebra so I will introduce s of g which is the algebra of polynomials on g star  and i of g which are the invariant polynomials on g star so the group the group there is an action  let's say of the of the group or of the Lie algebra on these polynomials and look at these invariants  so the Duflo isomorphism tells you that these two algebras are  properly isomorphic so the isomorphism is not revealed you cannot guess what it is just by  looking at these two algebras that it tells you that there is a canonical isomorphism  of these two algebras there's a center of the Lie algebra is indeed an algebra of polynomials  on g star so in particular for instance when you want to look at what corresponds to the  casimir cg in this algebra i of g that's minus b star plus b star of rho g rho g b star being the  du of b that is it's a polynomial of degree two on g star so center and enveloping algebras  so and I take l to be an element of the center z of g and I take its counterpart in the algebra of  polynomials i of g so what I'm saying is that this element 2d minus one of e this polynomial  restricts to an element of i of z of gamma so the restriction is not entirely is not entirely  natural but there is a natural projection from the Lie algebra on z of gamma so that you can define  in some sense the proper sense the restriction of this element so from a polynomial you get a  polynomial on the Lie algebra of the center and again from this polynomial now you rebuild a  differential operator you rebuild a differential operator on z of gamma with constant coefficients  so in other words you take the element of the center you look at the corresponding polynomial  you restrict it to z of gamma and you rebuild a differential operator acting on z of gamma  with constant coefficients I would say even acting on the complexification of z of gamma  sorry the center yes z of gamma is just a Lie algebra z of the centralizer of gamma  yeah the Lie algebra okay so the formula for general orbital integrals now instead of writing  it for the heat kernel I take any I take any kernel in some sense of the Kazimier so this is  a formula that I proved with Shushan in 2019 which says that the orbital integral of l so in other  words a new ingredient now is this l this element of the center which actually probably involves  all the possible Kazimiers that you were mentioning times a kernel an arbitrary kernel of the  of the Kazimier well I mean it's given by the action of a new kernel you take a new kernel  which now acts on the Lie algebra on i times k of gamma okay so you need to introduce to to  understand this this formula is not too complicated you you have the action on the right hand side  of the differential operator l z of gamma which acts on z of gamma and you compose it  with a kernel of the corresponding Laplacian on z of gamma and you make it act on this  distribution j gamma times in the case we introduce a twisting bundle the corresponding  character for the twisting bundle we evaluate this kernel at zero so in other words you make  it act on this distribution you have a smooth kernel you make it act on the distribution  we evaluate it at zero and the formula is that the original orbital integral is equal to this  new evaluation of smooth kernels on the Lie algebra of z of gamma okay so for l equals one  that's essentially the original formula so the principle of the proof we start from the formula  for this l equals one so when gamma is regular that is when it's regular that is if you like when  all its eigenvalues are distinct so it centralizes a carton subgroup the formula I gave before  for the heat kernel simplifies dramatically and can be differentiated in gamma and then ultimately  we use fundamental results of a rich chandra to evaluate the action of a differential operator  on our original formula the formula without the L to obtain a formula for l for gamma regular  and so if gamma is semi-sibler or bitery we use a previous result for gamma regular and  limit results of a rich chandra to obtain the general formula so let me just remind you what  the function j gamma is so therefore general semi-simple element gamma that sort of quotients  of determinants evaluated in the p and k part of the Lie algebra of z of gamma. Now, it  turns out that when gamma is regular, they are sort of cancellations, I would say supersymmetric  cancellations between the contributions of p and the contributions of k, so that the formula  simplifies thematically. So, more precisely, you can re-express the function of j of gamma in terms  of the roots of root system, but essentially of the purely imaginary roots of the reductive group.  So, the imaginary roots are the ones we just correspond to the action of k on the p and k  part, so that means that there is exactly as before where the eigenvalues were disappearing,  here is the p and k factors which cancel each other. And so, this allows us to treat in some  sense to prove that this function in the proper sense is smooth as a function of gamma, as long  as gamma is regular, that is, if it's eigenvalues, if you like, are distinct. So, I will complete this  by reminding you something that you all know, that the Langevin equation that in 1908 Langevin  introduced the Langevin equation mx double dot equal minus x dot plus u double u dot to reconcile  Bröhnian motion and classical mechanics. So, Bröhnian motion for m equals zero and classical  mechanics for m equals infinity. So, in this theory of the interpolating hyperliptical  appellation, m equals b square plays a role of a mass. So, this, I polyp, interpolation  plays exactly, in some sense, uses the tools, all the objects introduced by Langevin. At the time,  of course, there is no group theoretic interpretation. So, here is the, is a paper by Langevin in 1908  where you find exactly the equation written either what it before and with an interpretation that  you all know. Okay, I think I will just give the references here and happy birthday, Samsung.  Thank you very much. Yes? Yeah, I always have that main question about the constants Dirac.  Yes. So, like if you took some other Dirac on the group, what would go wrong or what would go right?  I mean, you see, there is the key thing which I have not explained is that the Kazimir lies in the center.  Okay, so in the equality of the trace is equal to the super trace, the fact that it lies in the center  plays a fundamental role because basically you need to prove that, you know, to write the super trace  of some super commutator vanishes and the fact that the bad term disappears is related to the fact  that it is a Kazimir. So, there is only one operator which will do what you want. All the other  ones will always contain something which makes them outside of the center. That's somewhat complicated  whatever book them. Yes, yes, yes. So, that's why you need absolutely the costum. Okay, so that's,  I did not explain this, but the Kazimir is in the center. I mean, and that plays a fundamental  role in the fact that the deformation is, yes? Two questions. Could you give an example of something  that, you know, businesses like myself can understand that you can do with these wonderful  formulas? Yes. And then the application, that's question one, question two is if you think of G  as being, for example, the group of gauge transformations, can you use this in the  DRS sphere? Yes, okay, so let me try to answer to this. Okay, so first of all about the role of  the judizic flaw. Why is your judizic flaw there? And then I will give an explanation, I think,  which is if you think of the judizic flaw and you think you write in coordinates, why do you have  a value over dx? If you take a symbol that you should make a Fourier transform with the x variable,  what you get is square root of minus one y xi. If you exponentiate this, you see that this is  Fourier transform. Okay, so the ultimate reason why the judizic flaw appears, it's because in some  sense it does for us mathematicians what it does is a Fourier transform. Okay, so we are  geometrically trying to implement the idea of the Fourier transform in a context of course. If you  think of Focke-Planck operators, I think it's fair to say nobody thinks of Focke-Planck operator of  doing Fourier transform of anything. Okay, it is a dynamical, statistical, physics system that you  try to describe, but the way we use it is this idea of Fourier transform. The second thing is that  in the case where the face is flat, that is where you work in the case of r or s1, everything can be  can be worked out explicitly in some sense. It is not so simple even that case to understand  mathematically what happens. So let me try to say it in words. So you have the question that you  know that you can quantize the group of simplectomorphisms. We can quantize it as a quantization to  operators. You pass from simplectomorphism to unitary operators. Okay, let me say it roughly.  What happens in this theory is that what you're quantizing is not simplectomorphism real ones,  you're quantizing complex ones. Okay, so the quantization of complex simplectomorphism  introduces a face. I mean if you look at the corresponding operator which appears on the other  side, it will no longer be unitary, but the worst is that it will be badly unbounded. Okay,  okay. However, if the consequences of this theory are true, that would mean that two operators which  look very different are ultimately the same because they are conjugate by something. Okay,  this something does not exist because it doesn't act, but still the operators behave  behave properly. That is, they still have the same spectrum. Okay, so if you were doing this  forgetting about this, you would just remain in the realm of classical Schrodinger-like  operators. If you just quantize these non-complex transformations, you pass to a completely different  category of operators. Yeah, so I think that the best explanation I can give for the moment.  Other questions?  Cannot put in certain functions, representations. Yes, yes, yes, you can put representation and  that's why you have this extra contribution of pairing of the a-roof function. I mean,  with a churn character function, which in this case is just the character corresponding to this  representation. Yes, so that's why it was there. Yes. Other questions? If not, thanks again. Okay, thank you.
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So it's a great honor and pleasure for me to come here for Samson's birthday. I mean we have been friends for how many years? I don't even know. 28. 28, more than that, no? 29. Maybe, okay. So I thought that it was a whole life anyway. So we have discussed about many things. I have listened to him on physics many, many times without being sure of understanding what he was saying. But I mean I sadly remember that when he talks to me he says, you're the unusual Frenchman. And I must say, I won't tell you what he refers to, but I'm very proud of this. Okay, so I will talk about Focke-Planck operators in the center of the enveloping algebra. So basically I mean I will explain certain things which are from physics point of view rather down to us, but maybe with applications to questions in mathematics which are of some interest. So I will basically explain how classical objects like cluplations can be deformed naturally to another category of operators which are in some sense operators well known in statistical physics like Focke-Planck operators. And this deformation is legitimacy. I mean it's not, you know, it's not for my own pleasure, but it was to connect with things like Zellberg's trace formula by an interpolation process. So the interpolation process I refer to is interpolation between the Laplacian on one hand and a dynamical system like the Joujizic flow on the other hand. And Zellberg's trace formula is exactly a sort of formula in which on one hand you have spectral data like Laplacian's and on the other hand you have Joujizic's and we will see our certain understanding of this formula will lead to explicit evaluations of objects which were not known before but using instruments which are in some sense taken from physics. So let me first of all start with extremely mental question, I mean considerations. So first of all the Euler characteristic, so you take X to be a compact remanian manifold and you know that the Euler characteristic is a global invariant that the alternates some of the dimensions of the homology groups of X. Now if you take diffeomorphism of X so it acts naturally on the homology so there is an action, linear action and so you can find the left set number which is again a global invariant which generalizes the Euler characteristic and which is just the alternates some of the traces of the diffeomorphism group acting on the homology. Now there is a way to compute explicitly this left set formula which is the so called fixed point formula or in the case of the identity Chern-Garbzman theorem but which is always based on the fact, I mean can be based on the fact that you can actually to compute the homology you can just use the Durand complex which homology is just the homology of X and if you introduce a remanian metric you introduce a former adjoint of the Durand operator and dx which is the little dx plus little dx star to be the corresponding drug operator and it's query is a hard-to-laplation. So basically the starting point of this sort of computation is to say that you can re-express left set number which is a global simple in some sense invariant in terms of super traces of operators which are now calculated on an infinite dimensional object and that the super trace of G exponential minus S times the Laplacian and this formula is true for any S. So in some sense if you just look at the right hand side there are many eigenvalues the contribution of zero eigenvalues of nonzero eigenvalues disappear which explains why the right hand side does not depend on S so that by making S tend to zero you obtain I mean you can obtain at least that the way of proving the left sets formula in terms of integral on fixed point set XG of the ailer form of TXG so if the points are isolated you just get a sum of plus or minus one. So let me now move to the real subject of my talk so I will consider now the heat operator on the manifold on the remanian compact remanian manifold X and I introduce again the Laplacian delta X and the corresponding heat current so now G for T positive it's the heat kernel at time T, T will be fixed so this is an operator which acts on smooth functions on X with values in R and so I will ask for rhetorical questions. So the first question is is the trace of the heat kernel a knowledge characteristic so can I think of this trace exactly in the way I was thinking of the trace of the difform of this before can I view this as sort of topological invariant so is this a generalize characteristic so the second question I will ask is exactly as before when I started from the chronology and I replaced the chronology by the Durand complex can I replace the smooth functions by a much bigger space by a much bigger space R such that I would write exactly formula of the same time as before re-expressing the trace of G in terms of a super trace of G but acting on a much bigger space equipped with a generalized Dirac operator which is exactly the analog of the operator DX which I introduced before and such that in a certain way I will instead of getting let's say churn Gauss-Bornay or the fixed point formulas when making B tend to infinity I would obtain something like Zellberg's trace formula which is an extension of Poisson formula so basically the difference with what I did before is now original space which was the chronology in the first example is now the space of smooth functions. So let me just formalize the analogy again so in the case of left shat number I had L of G which is left shat number generalized Euler characteristic on the right hand side I had a local formula for it which is a fixed point formula or churn Gauss-Bornay which is local and to prove the formula we use an interpolation involving the Durand complex and basically the interpolation just tells you that left hand side is ultimately equal to the right hand side. So this is exactly what we want to do in some sense I start on the trace of the heat kernel which is a global which is a global object and now of course I know more or less what I want to get on the right hand side which is Zellberg's trace formula but I want to find something in the middle which will do for us exactly what the Durand complex was doing before. So we are forced in some sense to consider the space of smooth functions in a manifold as the chronology of something the question is to know what is it the chronology of. So before going on but let me just remind you what Zellberg's trace formula is and a concrete example so this was the original formula proved by Zellberg so if X is a remand surface of constant scalar curvature minus 2 we introduce the closed udizix and their lengths L gamma. So Zellberg's trace formula in this case is an extension of Poisson formula so it expresses it expresses on the left hand side the trace of the heat kernel in terms of a bunch of terms the first term is a complicated one with an integral of r and with a volume factor in front of it the volume of x or it's not sigma is x and then there are a number of terms which are indexed by the closed udizix and with complicated factors so you see exponential minus a length square the usual thing in the denominator you have two hyperbolic sign of length square over two and then you have this infinite sum so in some sense what I will show is exactly finding what is a proper analytic and geometric structure which will interpolate between the two sides. So let me now move to the context on which I will effectively develop this theory so this will be the context of the symmetric spaces so I introduce G to be a real reductive group so if you don't know what reductive groups are just think of SL2R if you wish and k a maximal compact subgroup which is the case of SL2Y that's one and x equal g mod k is the corresponding symmetric space so I also introduce the carton splitting of the Lie algebra of the group so so so typically the Lie algebra splits into two pieces p and k there is a natural quadratic form of the Lie algebra which is positive on p and negative on k that's an orthogonal splitting and so this splitting of Lie algebras descends to a bundle of Lie algebras on the symmetric space Tx plus n modeled on the splitting g equal p plus k so again the basic example is g equal SL2R k it's just a matrix orthogonal matrix 2 by 2 matrix x is just a complex upper half plane and Tx plus n there is this n is a bundle of dimensions 3 over x so the analysis that we shall do will be done on the product of the group by its Lie algebra and quotient by k so quotient by k k acts on the group on the right it acts on the Lie algebra on the left and there is a standard procedure to do the quotient so the the analysis will be done on this and this subject is itself a geometric object so it is at the same time a group theoretic object but that's also a geometric object that the total space x x plus or hex hat of Tx plus n over x equal g mod k so you have this if you like the upper half plane you add this three dimensional bundle and so in the case of SL2 you will do the analysis I mean differential operators will occur on a space of total dimension 2 plus 3 which is 5 yes how is it the total bundle is trivial right the total bundle is trivial yes I mean the fact the bundle is trivial doesn't mean that the theory I will I mean it doesn't imply at least directly that the theory I will develop is trivial because the no but I mean I want to make the point that there will be a non-trivial coupling between the base and the fiber okay so we will do two separate constructions first of all the group and on the other side on the other hand on the Lie algebra so for the moment we forget about the quotient and only explain the two constructions on one hand on the group and on the other side on the Lie algebra so on the group side Kazimir and Kostard so if you have a reductive group like an acompactly group you have I mean you have the the the extension of the of the of the Laplacian or minus the Laplacian I would say I make it with a minus sign which is the Kazimir operator so that's an invariant differential operator on the group that takes into account the fact that there is this killing form which is positive in one direction negative on the other direction so that's why cg equal minus sum of e star i e i it takes into account the positivity or negativity of the form of the killing form B on the on the Lie algebra so if you like that an extension of Laplacian but that's a good Laplacian the p direction and a bad Laplacian with the wrong side in the k direction so that's why there are many Kazimir this is quadratic this is a quadratic Kazimir yes yes I will talk after all about all the possible Kazimir yes no I mean that will be the end of the talk okay so I introduce also the Clifford algebra of G equipped with a with an invariant form minus B and this Clifford algebra acts on the exterior algebra of G star so basically costant introduce an operator which whose square essentially is the is is the Kazimir okay so to do this I just introduce now the enveloping algebra U of G that the algebra of left invariant differential operators on the group so if you take the Lie algebra defines invariant vector fields and you look at the algebra differential operators generated by these vector fields at U of G and so the Dirac operator is a first order differential operator on the group and it can be described as a section of the Clifford algebra tensed by the by U of G is actually a differential operator order one of order one and it has first piece which looks like a Dirac operator the sum of c hat of E I star E I that's in some sense the classical Dirac operator and there is a correction which is a cubic correction of lengths three in the Clifford algebra and which involves the fundamental three form three form anti-symmetric three form on the Lie algebra which is given by B of U bracket VW that's an anti-symmetric three form and so that's the way that costant constructed is Dirac operator and so the formula of costant I remember that that that I was told by you that it was already proved by physicists before but that the square of the Dirac operator of costant is essentially up to a constant given by minus a Kazimier so this costant operator I mean doesn't act on smooth functions while the Kazimier act on smooth real functions the Kazimier acts on smooth function on the group with values in the exterior algebra of G star because of the Clifford algebra while Cg itself can be made to act on the smooth functions so this is such an identity we will use partly from an analytic point of view that is effectively try to reveal the hidden exterior algebra of G star now let me explain what we shall do on the Lie algebra side I told you the analysis is done on two pieces the group on one hand the Lie algebra and then we will couple them so on the Lie algebra G first of all we will make a weak rotation on the Lie algebra okay the Lie algebra G as I told you is not an Euclidean vector space it's a Lorentzian like vector space so we make a weak rotation to transform it into an Euclidean vector space so we replace p plus k by p plus i k and I introduced the harmonic oscillator on this gi so as you know the harmonic oscillator is given by this formula here where y is the generic is a generic point in gi and we will in some sense also on this gi introduce a Dirac operator a Dirac operator which is a Witten operator I mean that's that's connected with the Witten's idea of the twisting of the Dirac operator on a Euclidean vector space so we twist this on this Euclidean vector space the Dirac operator by the Gaussian by the Gaussian function so we get two new operators the modified Dirac which is d plus y wedge and the other one which is its swamble joint and these two operators also act now on smooth swarms on the Lie on on g whisk coefficients in the exterior algebra and so this operator here is a Dirac like operator acting on gi the sum of these two and the corresponding Laplacian that was a formula originally obtained by Witten tells you that up to a factor this Laplacian is given by the harmonic oscillator which is sort of bosonic number operator plus the fremianic number operator acting on the Lie algebra so we're going to couple these two constructions on one hand the costant operator acting on the group and on the other hand the Witten like operator acting on the Lie algebra so I will consider to construct now a new Dirac operator db parameterized by b positive so this operator db will be a combination of the Dirac operators on the group and on the Lie algebra so what does this mean this means that these new operator db will act on smooth functions on the group times the Lie algebra whisk coefficients in the exterior algebra of this time so remember I told you that the costant operator who is acting on functions on the group whiz values in the exterior algebra the Witten operator acts on functions on the Lie algebra whiz values in the same Lie algebra up to complexification so we can effectively combine the two so let me just give the formula so you see forget about the red factor so the first term in these db that's a new Dirac operator acting on a big space the first term is a costant Dirac operator and the second one is just a scaled version of the Witten like Dirac operator and there is a third mysterious term which is in red which is non-linear which is quadratic and whiz I mean whiz presence plays an essential role because if it was not there I would not be there either and I mean it's introduced it has to be forced because we will do the quotient thing by K remember that ultimately we'll descend everything to the quotient by K so that's this operator db so db is a K invariant operator it's not yes these whiz now correspond to the points in both directions separately or sorry sorry excuse me these whiz correspond to some points in the yes yes yes yes yes yes absolutely yes yes so in the gi direction you have two components yp and yk and that's why you take you take the league brackets okay so so that's exactly this so the the introduction of the quadratic term I won't have time to explain much about this is related you have to introduce these because you will eventually quotient things by K which introduces non-linearities by itself so what is this object that I will call the hyper elliptical aplation I will explain later what is hyper elliptic so it is just the operator lb you which is one half of db square minus dh d hat k square so remember that up to the quadratic factor I won't mention db is just a sum of the Dirac constant operator on g plus the written operator on the l algebra you take the square of the sum and you subtract in some sense the square of the first term so you undo what you had done except that you have less to is when you have a plus b square that a square plus ab plus ba plus b square so you subtract a square and you ultimately quotient the construction by K because all these subjects are K invariant they're not gene variant they are K invariant so as I mentioned before the le algebra descends to a vector bundle tx plus n which is sort of canonically trivial but not trivial as in the cladian vector bundle so let me introduce again the space x hat that's a bundle over x over the symmetric space whose fibers are tx plus n and so our operator lbx I explained before it will act on smooth sections over x script of the pullback of the exterior algebra of t star x plus n star so in other words g star is now replaced by t star x plus n star so using the bagman isomorphism I could put things in a more symmetric way instead of viewing things as a geometric object acting on this bigger space you could say I don't make the space bigger I just keep the space x down here but the price to pay is that you'll act on an infinite dimensional vector bundle involving the symmetric algebra which appeared also in untau minas talk this morning so it's the same it's the same object except I just have one copied set of having an infinite number of copies so let me give you first of all formula for this operator lbx so this operator lbx is supposed to be a deformation of the original scalar laplacian okay so this operator lbx if you just look at the red pieces it's essentially a forker plank operator so in other words this operator lbx it's harmonic oscillator in the fibers of tx plus n which I view as an eclidian vector bundle after weak rotation there is this harmonic oscillator in red and there is this piece here which is what I call the judizic flaw well if there was only tx if there was only tx you have a remanded manifold x you have the you have the fiber tx and on this big object of dimension twice the original dimension you have the judizic flaw and that's what's represented by the nabla y tx so if you forget about the other geometric terms we just have essentially a forker plank operator which is a coupling of this harmonic oscillator in the fiber and the judizic flaw in some sense which is a vector field on the base so this operator is non-self adjoint I mean it is not self adjoint obviously not the judizic flaw is anti symmetric it is it is not elliptic elliptic means that there is I mean it just differentiates once in the base direction not twice so it doesn't it's not a laplacian in the usual sense but what I'm saying is that from an analytic part of you it's a very good operator it's almost as good as the laplacian it's why it's called hyperliptic and what I'm saying besides is that this operator deforms naturally the original laplacian so this actually you can prove in a rigorous mathematical way this operator lbx deforms in some sense the original casimir operator or laplacian if you like acting on the base x so that means that when b10 to 0 there is a collapsing of the space x script to the space x so let's try to understand a little bit this collapsing I mean what happens is that when b10 to 0 the harmonic oscillator gets scaled by a big factor and that in some sense when b10 to 0 you ignore everything of the eigen eigen spaces or eigen values of this operator except the zero eigen value that's exactly a collapsing phenomenon which is in some sense exactly the same sort of collapsing that you have when you have look at calusacline theory in which you have very small circle the circle gets smaller and smaller because I mean in some sense you just concentrate on the fundamental on the ground state of the circle you have exactly the same phenomenon here you concentrate on the ground state of the harmonic oscillator except that there is here a non trivial coupling between the fiber and the base via the judizic floor which ultimately explains I won't prove that that in the proper sense the operator lbx deforms actually when b10 to 0 there is a language there is a theory in which you can say this is transport focal plan operator deforms the original laplacian but it is not only that it deforms it because it still captures a lot of information on the original laplacian in particular on its spectrum as long as it will work with compact quotients so as b10 to infinity after rescaling you see that it is the judizic floor which dominates so when b10 to 0 deforms to the laplacian when b10 to infinity the judizic floor dominates and if you think of taking traces of heat kernels you can see why when b10 to infinity you will ultimately select closed judizics so let me apply these two locally symmetric spaces now so original symmetric space x is just rn as a smooth space is just rn so we need to build first of all a compact manifold from the space so basically what you do is that you take a discrete subgroup of the group g and gamma and I will assume that it is co-compact and torsion free so this is just a complicated language to say that gamma acts freely on the in a discrete way on x so you can form the compact quotient gamma on x that's a compact manifold exactly like s1 is a quotient of r by the action of z and so I will give my first identity the first identity says that the trace on the smooth functions on z forget about this e just take e equals r for the moment on smooth functions of the heat kernel is equal to the super trace of the hyperliptic heat cut so we have exactly an identity which is similar to what I wanted to have before to deform the evaluation of the heat kernel into something more complicated involving more coordinates and actually if we think of it this idea of resolving of replacing this of saying the space of smooth functions is the cormology of something is actually embedded in the construction I gave before through this written harmonic oscillator I won't give details on that but we have exactly what we want sorry what is small c oh c is just the b square I mean that the defect in in in in in in costant's formula that just a defect in costant's formula that the lengths of rho g that's a square excuse me question about this this theorem yes so isn't that like the in physics you would say that you have a super symmetric s l2 sigma model and this is just localization yes so this is exactly so we have I mean in some sense we have constructed a super symmetry in a theory which is not super symmetric so that was yeah that's early algebra the algebra is where the fermions are and this is just localization the standard I mean localization seems not no I mean you you just we will localize like to rock okay we'll take b we'll take b b b tend to be busy and then we will be in it and then you get your physics yes yes that's so this is what we will do but sorry wasn't that done in the physics literature long ago no I don't think no not for that no no because I mean the idea that focal plan cooperators deform the Laplacian you see what this contains in particular is that the spectrum of the Laplacian is contained in the spectrum of these focal plan cooperators okay these focal plan cooperators have some real eigenvalues they have non real eigenvalues so ultimately when taking the super trace the non real eigenvalues disappear and what's left is just a real eigenvalues so you have had more eigenvalues but no okay so let me now say that there is a super symmetric interpretation involving involving the complication the complexification of the group but I won't mention it yet so splitting the identity so each side I mean since you're on a compact on a copent quotient by a discrete group each side splits as an infinite sum indexed by the conjugacy classes and gamma so this I mean exactly like Poisson original formula says that you take kernel on s1 is just a sum over z you can exactly do the same thing and now the miracle is that this identity splits as an identity that is you can forget about the group gamma and you could say the identity remains true in some sense element of gamma by element of gamma it is not an identity of sums the identity remains true and that's an identity of objects which are called orbital integrals so let me briefly mention what semi-simple orbital integrals are orbital integrals are fundamental in in in the theory of representation automorphic forms and so on so let me just take the simplest case actually I take a semi-simple element gamma and I call bracket gamma its conjugacy class so basically an orbital integrals that's its name is that you integrate the heat kernel on x I mean you need to introduce some notation for that on the orbit on the adjoint orbit of gamma by the action of g on the element gamma and the integral is an integral of the quotient of the quotient by z of gamma so these are the objects which are the ingredients which appear when you express the trace of the compact quotient in terms of the group gamma in some sense you re-express this trace as objects which are the orbital integrals so let me give a geometric description brief geometric description of what the orbital integrals are so I will just introduce this picture so this this is a flat space which is supposed to represent the symmetric space x equals rn if you like so inside this rn or inside the symmetric space there is a sub manifold which is x of gamma which is also a symmetric space and it can be geometrically constructed by minimizing by minimizing the so-called displacement function d gamma of x which is d of x and gamma x where d has a remanian distance now the remanian distance on such symmetric spaces is convex in the sense that it is convex on judizek so the minimizing set is non-empty because gamma is semi-simple and it is convex it's x of gamma the x of gamma is actually a symmetric space for the centralizer of gamma so how can you understand what the orbital integral is I just describe it geometrically what you do is that you take the heat kernel you take the heat kernel you take a normal you take an x0 in x of gamma you take a normal and you take a generic point why you look at its image bar gamma gamma is an azometry it maps normal judizek to a normal judizek and you find that because of negative curvature the mutual distance of y and gamma y grows at least linearly when y tends to infinity and you use ultimately the fact that the heat kernel decays at long distances like a Gaussian on the symmetric space so that the above integral exists so in other words you take the heat kernel at y and gamma y in integrated and normal directions you ultimately get a Gaussian integral which converges that's the orbital integral so the point of the hyperliptic theory will be to push ultimately what we will do is push the integral to x of gamma to the closed judizeks so the second fundamental identity is that effectively instead of having an identity of traces you get an identity of the so-called orbital integrals which are now calculated using heat kernels over this symmetric space not over the compact coefficient this is true for any b so let me now take the limit as b tends to infinity so after rescaling this operator lb the original operator we had when b tends to infinity is dominated in a way by the judizek flaw except that there is this extremely large term quartic term which dominates everything which means that localization will take place on commuting pairs yn and ytx and so ultimately the computation of the orbital integral localizes near the manifold of judizeks this x of gamma associated with gamma as I explained before the point of this is to push the integral to x of gamma so let me try to explain what is the structure of the formula for these orbital integrals so I take a gamma to be a semi-simple element which means that it is in a certain way diagonalizable so I take its reduced form I write it as a product of a self-adjoint if you like a unitary element e to the a times k minus one where a is in p and k is in k so after conjugation you can always do that and these pairs are commuting so let me call z of gamma to be the centralize of gamma and with le algebra splits again as p and gamma plus k of gamma so z of gamma is still is still a semi-simple is still a reductive group sorry yes just on the previous slide yes so I think it's related to the previous awkward question when you take the p to infinity yes this corresponds to the quarter of the flows right yes the judizeks yes so essentially it constitutes localization but your result comes from the deformed l right yes essentially this is not this is different for your plan plus your so so the localization will come from this deformed focacplug because you're taking the trace yeah but this result is not from the like conventional focacplug this is already deformed one because your lb is defined as a constant squared minus constant yes but I told you the original result tells you that the original object you went to compute is equal to the deformed one okay this was the first thing the original object you went to compute is equal to the form one so if you localize a deformed one you get something which is equal to the original object okay so you started from the casimir you deform it to this focacplug like operator but still you preserve the original object which means if you were working on the compact ocean the exact reproducers in this limit the same the same contribution sorry how about that so essentially in this limit it reproduces original result yes yes so it means like your deformation and yes there is a deformation but you do not change anything okay and and and now of course the result I have is no longer spectral result okay it's no longer the point is of this thing is if you can analyze now things judizik if you like close judizik by close judizik instead of having to localize an infinite number of judizik at the same time can you make a comment about yes the relationship to the physics yes because there is this paper on localization that you do the same formula by kecky or cacchina yes yes so what's the relation to what you're doing I don't know the paper that you mentioned okay maybe we'll talk about this later okay so let me give the evaluation of the semisimple orbital integrals and this formula will extend the formula of zelberg I gave at the beginning so there is an explicit function j gamma which is a function on i k of gamma so let's fix the notation again you have gamma semisimple element you have its the algebra z of gamma which splits into p and k of gamma and so this function here is defined on i k of gamma the weak rotated i k of gamma such that the original object the original object which is a trace gamma okay the orbital integral associated with the elliptic heat kernel and now let's look at the formula the formula should be looked at as an a tier singer like of index theorem in which integration on the manifold is replaced by integration on i k of gamma you have the first term which is j of gamma which will be an analog of the Todd or a roof genius but now it's a function now I have a churn character like if I need not introduce a twisting bundle so if you like you can ignore it if you're interested in sections of vector bundles you have to introduce it and you have a Gaussian integral here and so really the mystery lies in this function j of gamma so I told you it's a function it's not a characteristic class it's a function and so this function j of gamma it looks intimidating but in some sense it's a super symmetric version of the a roof genius on this manifold x you had these two bundles tx plus n but actually from the point of view of this point of view you have to look at this as tx minus m so the contribution will be the ratio of the corresponding a roof gene genera or a roof functions so I give the function here I give the formula which is every time a ratio of functions evaluated on the p and the k part the p part always appears in numerator the k part in denominator and these functions take into account the fine splitting of the Lie algebra under the action under the adjoint action of the representation of k of gamma the detail for the moment is not important here so the center of the enveloping algebra so again let me remind you that u of g is the enveloping algebra that's a non commutative algebra and inside there is a center which is a commutative algebra so I will introduce s of g which is the algebra of polynomials on g star and i of g which are the invariant polynomials on g star so the group the group there is an action let's say of the of the group or of the Lie algebra on these polynomials and look at these invariants so the Duflo isomorphism tells you that these two algebras are properly isomorphic so the isomorphism is not revealed you cannot guess what it is just by looking at these two algebras that it tells you that there is a canonical isomorphism of these two algebras there's a center of the Lie algebra is indeed an algebra of polynomials on g star so in particular for instance when you want to look at what corresponds to the casimir cg in this algebra i of g that's minus b star plus b star of rho g rho g b star being the du of b that is it's a polynomial of degree two on g star so center and enveloping algebras so and I take l to be an element of the center z of g and I take its counterpart in the algebra of polynomials i of g so what I'm saying is that this element 2d minus one of e this polynomial restricts to an element of i of z of gamma so the restriction is not entirely is not entirely natural but there is a natural projection from the Lie algebra on z of gamma so that you can define in some sense the proper sense the restriction of this element so from a polynomial you get a polynomial on the Lie algebra of the center and again from this polynomial now you rebuild a differential operator you rebuild a differential operator on z of gamma with constant coefficients so in other words you take the element of the center you look at the corresponding polynomial you restrict it to z of gamma and you rebuild a differential operator acting on z of gamma with constant coefficients I would say even acting on the complexification of z of gamma sorry the center yes z of gamma is just a Lie algebra z of the centralizer of gamma yeah the Lie algebra okay so the formula for general orbital integrals now instead of writing it for the heat kernel I take any I take any kernel in some sense of the Kazimier so this is a formula that I proved with Shushan in 2019 which says that the orbital integral of l so in other words a new ingredient now is this l this element of the center which actually probably involves all the possible Kazimiers that you were mentioning times a kernel an arbitrary kernel of the of the Kazimier well I mean it's given by the action of a new kernel you take a new kernel which now acts on the Lie algebra on i times k of gamma okay so you need to introduce to to understand this this formula is not too complicated you you have the action on the right hand side of the differential operator l z of gamma which acts on z of gamma and you compose it with a kernel of the corresponding Laplacian on z of gamma and you make it act on this distribution j gamma times in the case we introduce a twisting bundle the corresponding character for the twisting bundle we evaluate this kernel at zero so in other words you make it act on this distribution you have a smooth kernel you make it act on the distribution we evaluate it at zero and the formula is that the original orbital integral is equal to this new evaluation of smooth kernels on the Lie algebra of z of gamma okay so for l equals one that's essentially the original formula so the principle of the proof we start from the formula for this l equals one so when gamma is regular that is when it's regular that is if you like when all its eigenvalues are distinct so it centralizes a carton subgroup the formula I gave before for the heat kernel simplifies dramatically and can be differentiated in gamma and then ultimately we use fundamental results of a rich chandra to evaluate the action of a differential operator on our original formula the formula without the L to obtain a formula for l for gamma regular and so if gamma is semi-sibler or bitery we use a previous result for gamma regular and limit results of a rich chandra to obtain the general formula so let me just remind you what the function j gamma is so therefore general semi-simple element gamma that sort of quotients of determinants evaluated in the p and k part of the Lie algebra of z of gamma. Now, it turns out that when gamma is regular, they are sort of cancellations, I would say supersymmetric cancellations between the contributions of p and the contributions of k, so that the formula simplifies thematically. So, more precisely, you can re-express the function of j of gamma in terms of the roots of root system, but essentially of the purely imaginary roots of the reductive group. So, the imaginary roots are the ones we just correspond to the action of k on the p and k part, so that means that there is exactly as before where the eigenvalues were disappearing, here is the p and k factors which cancel each other. And so, this allows us to treat in some sense to prove that this function in the proper sense is smooth as a function of gamma, as long as gamma is regular, that is, if it's eigenvalues, if you like, are distinct. So, I will complete this by reminding you something that you all know, that the Langevin equation that in 1908 Langevin introduced the Langevin equation mx double dot equal minus x dot plus u double u dot to reconcile Bröhnian motion and classical mechanics. So, Bröhnian motion for m equals zero and classical mechanics for m equals infinity. So, in this theory of the interpolating hyperliptical appellation, m equals b square plays a role of a mass. So, this, I polyp, interpolation plays exactly, in some sense, uses the tools, all the objects introduced by Langevin. At the time, of course, there is no group theoretic interpretation. So, here is the, is a paper by Langevin in 1908 where you find exactly the equation written either what it before and with an interpretation that you all know. Okay, I think I will just give the references here and happy birthday, Samsung. Thank you very much. Yes? Yeah, I always have that main question about the constants Dirac. Yes. So, like if you took some other Dirac on the group, what would go wrong or what would go right? I mean, you see, there is the key thing which I have not explained is that the Kazimir lies in the center. Okay, so in the equality of the trace is equal to the super trace, the fact that it lies in the center plays a fundamental role because basically you need to prove that, you know, to write the super trace of some super commutator vanishes and the fact that the bad term disappears is related to the fact that it is a Kazimir. So, there is only one operator which will do what you want. All the other ones will always contain something which makes them outside of the center. That's somewhat complicated whatever book them. Yes, yes, yes. So, that's why you need absolutely the costum. Okay, so that's, I did not explain this, but the Kazimir is in the center. I mean, and that plays a fundamental role in the fact that the deformation is, yes? Two questions. Could you give an example of something that, you know, businesses like myself can understand that you can do with these wonderful formulas? Yes. And then the application, that's question one, question two is if you think of G as being, for example, the group of gauge transformations, can you use this in the DRS sphere? Yes, okay, so let me try to answer to this. Okay, so first of all about the role of the judizic flaw. Why is your judizic flaw there? And then I will give an explanation, I think, which is if you think of the judizic flaw and you think you write in coordinates, why do you have a value over dx? If you take a symbol that you should make a Fourier transform with the x variable, what you get is square root of minus one y xi. If you exponentiate this, you see that this is Fourier transform. Okay, so the ultimate reason why the judizic flaw appears, it's because in some sense it does for us mathematicians what it does is a Fourier transform. Okay, so we are geometrically trying to implement the idea of the Fourier transform in a context of course. If you think of Focke-Planck operators, I think it's fair to say nobody thinks of Focke-Planck operator of doing Fourier transform of anything. Okay, it is a dynamical, statistical, physics system that you try to describe, but the way we use it is this idea of Fourier transform. The second thing is that in the case where the face is flat, that is where you work in the case of r or s1, everything can be can be worked out explicitly in some sense. It is not so simple even that case to understand mathematically what happens. So let me try to say it in words. So you have the question that you know that you can quantize the group of simplectomorphisms. We can quantize it as a quantization to operators. You pass from simplectomorphism to unitary operators. Okay, let me say it roughly. What happens in this theory is that what you're quantizing is not simplectomorphism real ones, you're quantizing complex ones. Okay, so the quantization of complex simplectomorphism introduces a face. I mean if you look at the corresponding operator which appears on the other side, it will no longer be unitary, but the worst is that it will be badly unbounded. Okay, okay. However, if the consequences of this theory are true, that would mean that two operators which look very different are ultimately the same because they are conjugate by something. Okay, this something does not exist because it doesn't act, but still the operators behave behave properly. That is, they still have the same spectrum. Okay, so if you were doing this forgetting about this, you would just remain in the realm of classical Schrodinger-like operators. If you just quantize these non-complex transformations, you pass to a completely different category of operators. Yeah, so I think that the best explanation I can give for the moment. Other questions? Cannot put in certain functions, representations. Yes, yes, yes, you can put representation and that's why you have this extra contribution of pairing of the a-roof function. I mean, with a churn character function, which in this case is just the character corresponding to this representation. Yes, so that's why it was there. Yes. Other questions? If not, thanks again. Okay, thank you.
Focke-Planck operators in the center of the how classical objects like cluplations can be Zellberg's trace formula by an interpolation formula will lead to explicit evaluations of formula we use an interpolation involving the the trace of the heat kernel in terms of a bunch this and this subject is itself a geometric Kostard so if you have a reductive group like an the extension of the of the of the Laplacian or square of the Dirac operator of costant is is its swamble joint and these two operators also and this formula will extend the formula of associated with the elliptic heat kernel and now that u of g is the enveloping algebra that's a the formula for general orbital integrals now