Hypoelliptic Laplacian and the trace formula
Bismut, Jean-Michel
The hypoelliptic Laplacian gives a natural interpolation between the Laplacian and the geodesic flow. This interpolation preserves important spectral quantities. I will explain its construction in the context of compact Lie groups: in this case, the hypoelliptic Laplacian is the analytic counterpart to localization in equivariant cohomology on the coadjoint orbits of loop groups. The construction for noncompact reductive groups ultimately produces a geometric formula for the semisimple orbital integrals, which are the key ingredient in Selberg trace formula. In both cases, the construction of the hypoelliptic Laplacian involves the Dirac operator of Kostant.
 It's going to be difficult. Maybe I'll redo it. OK, so on this left-hand side, you should think that there is an integral over the loop space. And on the right-hand side, just a sum over selected loops, which are precisely the closed-ruthy 6. And it is in this sense that eventually the interpretation of the proof and of the formula I will give will be sort of as a localization formula in equilibrium homology, where on the left-hand side, we have something evaluated on the full loop space, integral full loop space. And on the right-hand side, we evaluate simply sums over closed-ruthy 6. So that's a stupid question. But when you say equal, these things exactly equal. Exactly equal. And that's a funny expansion. No, no, it's not. It's not a funny expansion. In this case, it's exactly equal. It's as equal as it can be. It's an equality that we will eventually describe as an identity of the same sort as an index formula in some way. So that's an equality. So the purpose of the talk will be to extend the very explicit formula to a between locally symmetric spaces, explain how to obtain the formula by an interpolation process. So we will actually interpolate smoothly between the left-hand side and the right-hand side. That connects with infinite dimensional version of Deuysse Weifengmann-Berlin-Wang formulas. So first of all, left-sheds formulas. So I will start with some very general arguments. So Euler characteristics. So I take x to be a compact manifold. And I define the Euler characteristic. So as you know, there are many ways of computing the Euler characteristic, either through a triangulation, but there is another way through the heat kernel. So you introduce basically the Hodge-Loplaschen square x associated with a given remanian metric on the manifold. And we have this basic formula, which says that the Euler characteristic of x can be evaluated in terms of traces of certain heat kernels. Actually, there is a sign TRS, which means supertrace. It's graded trace. It's alternating sum of traces. And this is for the heat kernel of the Hodge-Loplaschen acting on the various forms of various degrees. So you take the alternating sum of these traces, and you get the Euler characteristic. This formula is quite easy to prove. So actually, one of the goals, one of the uses of such a formula, in this case, can sit in making s 10 to 0. And this way, you will obtain Schoen-Gausbohne. So more or less, this is a scheme of how to obtain the other heat equation all index formula. Now, left sets formula. So I take slightly more refined thing. I say g to be a diffeomorphism in diffeomorphism of x. And so I know that g acts on the quorumology of the manifold. So then you can define the left sets number. That's the alternating sum of the traces of the action of g on the quorumology. This is a global invariant. So again, you still have a formula of the type of McKinsey, which says that the left set number, which is something which only involves the quorumology, can be evaluated in much, with a much bigger object, which is the full-term uncompact, just as before. And the formula is almost the same, except that you introduce here the action of the diffeomorphism in the trace. And again, such formulas are quite easy to prove. So by making s then to 0, we obtain l of g as a sum of local sum of contributions of the fixed point of g. That's the left sets formula. So I wrote here an interpolation formula, in which here there is a g missing. So the interpolation here is meant to do the following thing. If you make s then to infinity here, you obviously get the left set number, because you obviously project on the harmonic forms, which represent the quorumology. And on the right-hand side, we make the limit as s then to 0. You obtain whatever formula there is for the left set number of g. So at the same time, this object remains constant and interpolates naturally between the left-hand side and the right-hand side. So this is exactly as the mental mechanism will apply in the case of the trace formula. So let me now take a compact manifold, x compact remanian manifold. Let me introduce delta x to be the Laplace-Baltrami operator on x. And then for fixed t, I just call g the heat current. So I deliberately use a segmentation that I used before for the nithiomorphism. So I'm going to ask four questions. So the first question is, is the trace of the heat kernel an Euler characteristic? What I mean by this is, can I think of the trace of the heat kernel as being an Euler characteristic of exactly the same type as before, just by doing the proper changes? Now, the second question is, can I express this Euler characteristic, or if you like, is generalizing the heat current using a much bigger object? So before, in the case of the chronology, we have the chronology, and we resolved it by the Durand complex. I'm asking, can I resolve in a certain way the space of smooth functions with values in R by a certain complex and re-express the trace here in terms of alternating some of traces on a much larger object? So the analogy again is passing from the chronology to the Durand complex. We want to pass here from the smooth functions to some complex which will resolve the smooth functions on X. And the third question will be, by making b tend to infinity, do we obtain Zellberg's trace formula? So in other words, can we think of the same type of trace formula as a generalized index formula? So again, I gave here the interpolation scheme. Here is a trace here on smooth functions of g of the heat kernel, which is a global invariant. By making b tend to infinity, we still don't know what the parameter is. We will obtain Zellberg's trace formula that we will think of as being a local formula. So what this means by analogy is that we will have to think of the smooth functions as the homology of some complex and even more as the harmonic objects in some complex. So I will try progressively to give the instruments which allow us to do this. So the answer is yes. So first of all, the question of resolving the state of smooth functions. So in the case of a manifold, again, we took the homology and we resolved it in the proper sense by the Durand complex. Here we're going to ask, can we think of the smooth functions on x as being the homology of something? So let me construct an infinite number of ways of doing this. So I just give myself a real vector bundle on x. And I call this script to be the total space of E. And then I introduce the space R. So this is the relative Durand complex. That the smooth forms are the total space, but which are just forms along the fiber. So this family of Durand complexes along the fiber. And DE is just a fiber-wise Durand complex. Now, I claim that my plurgery is lemma. The co-homology of this complex is precisely the smooth functions on the base. Well, this is because the co-homology of the Durand complex itself is made. Ultimately, it comes at a degree 0. It is a function which are constants, fiber-wise, meaning that they come from the base. So this is a complex whose co-homology is the smooth functions. Now, does this complex have a Hodge theory? Does this new complex R? Does it have a Hodge theory? So I claim that there is such a Hodge theory, because if you introduce GE to be an Euclidean metric, there is a Hodge theory for all these vector fibers. And this Hodge theory is in some sense given by the Wittner-Lapplation. So just a short reminder about Wittner-Lapplation. So in very simple case, I give myself E to be an Euclidean vector space. The classical obvious Hodge-Lapplation has continuous spectrum. But in some sense, the Wittner-Lapplation has discrete spectrum. So you can view it in one way, is to twist the Durand operator by the Gaussian function. Do not change the co-homology this way. So you construct the corresponding Hodge-Lapplation by taking the formula joint. What you get, essentially, is a Laplacian. You get a first operator, which is H, which is a harmonic oscillator, and N, which counts the degree in the exterior algebra. And so you recover, in some sense, the proper computation of the co-homology of a vector space, which is concentrated in degree 0, because H is the kernel of H is one dimensional, and N, 0 forms. So just to remark here that via the so-called Bergman's isomorphism, you can instead replace the Durand complex of E by the algebraic Durand complex on E. That is, this complex is quasi-isomorphic to this algebraic Durand complex. And then we see appearing here the polynomials, which are the symmetric algebra. And as we know, which is a natural inverse to the exterior algebra. So this construction here will play an important role in the cycle. So now we have our base manifold, X, and the fiber bundle, which is E. And the question will be to be how to couple the base and the fiber so that we can do something. So we need to introduce, first of all, the heat operator on the base, X, coupled with the written Laplacian on E. So this is the way that we'll approach the construction of this ultimately high-polyptical Laplacian, how to couple the base, which contains the heat kernel, with the fiber, which contains the written Laplacian. So let me explain the construction explicit in the case of a symmetric space. So let G be a real reductive group, K to be a maximal compact subgroup, and X equal G over K to be the symmetric space. I then introduce a corresponding carton splitting equipped with the non-generated bill in a form B. And this carton splitting descends to a vector bundle, E, which is made of two pieces, TX plus M. So it's a general fact that on a symmetric space or locally symmetric space, the tangent bundle has a natural auto complement, which is this thing which I call N. TX is modeled on B, and N is modeled on K. And actually, our interpolation will ultimately be built on the total space of such a vector space, of such a vector bundle. So we will explain how to couple X and TX plus M by lifting first everything to G multiplied by the realtor. So just an example in the case where G is a sum of R, K is S1, X is the hyper-alf-plane, and TX plus M is of dimension 3, so that the total dimension of fiber bundle is of dimension 5. So how are we going to construct this coupling? So Casimir and Kostalt. So first of all, on the reductive group, we have the Casimir operator, which is a second-order differential operator acting on the group, which is neither elliptic, neither hyperbolic. So it is, in some sense, positive in the p-direction and negative in the k-direction. That's a second-order differential operator, and which is in the center of the vector space. It is in the center of the enveloping algebra. Now I introduce the clefted algebra C hat of G, which is the clefted algebra of G minus B. And as we know, this clefted algebra acts on the exterior algebra of G stuff. So I also introduce U of G, which is the enveloping algebra. So if you like it, the algebra of differential operators of invariant differential operators of the group G, and the clefted Kostalt-deroc operator is a section of the clefted algebra tensed with the enveloping algebra. So it's a slight but significant modification of the construction of the classical deroc operator. It is constructed via the fundamental closed three-form of the group G, which I wrote here. And here is a short formula for it. You don't need to know this, so this is just, in some sense, the classical deroc operator, where the EIs are just differential operators acting on G. And here there is a cubic correction in terms of the clefted image of the three-form. Now the basic result of Kostalt is that the square of the Kostalt operator, with my conventions, is equal to minus the casimir. You could have another convention in which you would get the plus the casimir, plus some irrelevant constant. So what this means is that the casimir operator has a natural differential operator square root. And this we will exploit in the context of analysis. So let me observe that the Kostalt operator, what it acts upon, it acts on smooth functions on G with values in the exterior algebra of G star, while the casimir operator itself acts on the smooth functions on G with values in R. So in some sense we get a sort of straight root of the Laplacian, but it acts on the wrong space. So it's like straight root of minus one, which was just outside the real line. Kostalt, because I had K for, okay, I thought of this. I had K to be the maximal compact subgroup, and I did not want to have the question, what is K Kostalt? So this was made for you. Okay, so the operator Db now. So let me construct a sort of generalized Dirac operator. This Dirac operator will act on G product-ally algebra. So this new Dirac operator will act on smooth functions on G tensor, the types of the algebra, with values in lambda of G star. Okay, so, sorry, this is going too fast. So here is the formula. So I have a first term, which is just a Kostalt Dirac operator. So this one acts on the infinity of G tensor with lambda of G star. This is for the first piece. This one, you should not look at it. Absolutely not. Do as if it was not there. So this piece here is just a version of the Hodg-Durham written operator. That is, that's a sort of D plus D star, if you like, but in the proper Hodg theory. That is, in the proper Hodg theory of P and in the proper Hodg theory of K, which involves a written twist. Now you see that these two pieces, P and K, are treated differently. I mean, this different treatment reflects the fact that we are on a reductive group. But essentially, the effect when you square this is just to produce a very good Hodg-Lop-Lasch. So in some sense, these D papers, D per star, it acts on sin infinity of G of P, sorry, with coefficients in lambda of P star. And the other one acts on sin infinity of K with coefficients in lambda of K star. So if you combine all of these, you find that this new operator, D B, which I will call, for lack of a better name, a Dirac operator, acts on the smooth function of G cross G with values in lambda of G star. So what is the high-polyptical operation in this context? So what I'll do for the moment, I'm working on G, on the group G, 10 times the l-algebra G. So I take the square of the previous operator, D B square, and then I subtract from this square, the square of cos dot. So if you like, I do A plus B square minus A square. And here is a formula for this operator that I will explain, I will show to you. But before doing this, I descent, I quotient, the whole construction by K. So in other words, for the moment, we've done analysis on the group times the l-algebra, but then we have to descend everything to the symmetric space. So as I explained before, the l-algebra G descends to Tx plus m, G with its cut-out splitting, and G cross the l-algebra to the total space x hat of Tx plus m. So our new operator, lbx, which is obtained by descent, will act on the smooth functions on x hat, with values in the exterior algebra of T star x plus m star. So again, using the Fibromois-Bartmann isomorphism, instead of viewing the construction as acting on a bigger space, which is a total space of a vector bubble, you can instead view this operator as acting on an infinite-dimensional vector bubble. The infinite-dimensional vector bubble is just the symmetric algebra of T star x plus m star, multiplied by the exterior algebra of T star x plus m star. So here is the formula for this lbx. So this lbx contains along the fiber the harmonic oscillator of Tx plus m, but in some sense along the fiber it is just the written population. What we see appearing here as an operator which just differentiates in the base direction is actually the judizic flaw. And we have a bunch of other matrix terms that I will not explain. There is also this extra term here, which is a term of order four, which means that our operator, we are quitting the unshattered word of harmonic oscillators to go to something slightly more complicated of operators with coefficients quadratic, but this piece is absolutely necessary to make the theory work. So this operator is hyperliptic. So what does this mean hyperliptic here? That means that it does a good analysis in spite of the fact that it just differentiates twice in the fiber direction, but only differentiates once in the base direction. So analytically it is good, it is part of the class of operators concerned, considered by Hermandem. And so what I'm saying, and I will not show this, is that in the proper sense this operator deforms the original Casimir operator acting on X, or if you like deforms the original Laplacian. There is a collapsing phenomenon which is relatively subtle, which means that analytically splitting this operator deforms the original Laplacian. But now the wonderful thing is that when B tends to infinity, it will be the judizic flaw which will dominate. And so when B tends to infinity, this will force the localization on closed physics when you consider traces of it cones. So the case of locally symmetric spaces now, so let me take gamma to be a compact torsion free discrete subgroup of G, and then we consider the quotient of X by gamma, which is a compact locally symmetric space. So all the constructions I have done, so I have done constructions over G, then I descended these constructions over X, and now of course I can descend the construction to the quotient, because basically all the objects I introduced are properly equivariable. And so the fundamental identity is to say that the trace, in some sense of ordinary heat kernel, so Cz, in some sense you can think of this as a heat kernel, now Cz minus C, as being the, this is a sort of, I mean it's just modification of heat kernel by constant, is equal to the super trace of the corresponding hyperliptic operator. So exactly like an index theory where you say the original index, or Fshad number, is equal to the super trace of a heat operator acting on a bigger space, we have exactly the same sort of phenomenon here, and the proof is in this context rather elementary, it relies on the fact, on the collapsing of this operator to this one as B10 to zero, plus the invariance of the super trace under deformation by B, which is essentially an index theoretic proof. However this is not good enough for our purpose, because in some sense we're not going to apply to play the Zellberg's trace formula in its algebraic version. So actually when you have the trace of each kernel, locally symmetric space, you know that you can re-express it as a sum of objects which are evaluated on the symmetric space. So you can go back to the symmetric space. So in other words, in this equality, we can as well evaluate the left hand side and the right hand side as being sums, infinite sums, over the district root gamma. And now the true miracle is, is that it's not only true that the trace on the compact locally symmetric space is equal to the super trace again over the compact locally symmetric space, but orbital integral by orbital integral, there is a preservation. So now we're going to move to different sort of geometry. So the identity splits as an identity over orbital integrals, and this is why we will be able eventually to give geometric formula for that. So semi-simple orbital integrals. Let me take gamma to be semi-simple in G, and let me call gamma, right, it's country AC class in G. Then you can, this has already been introduced here, introduced in the proper sense, the orbital integral, in this case of the heat kernel, which is an integral of the quotient of G by the centralizer of gamma in the group. So in the proper sense, this integral makes sense, in spite of the fact that Pt of X is just a heat kernel of X over the symmetric space. So what I'm going to try to explain is how first of all to geometries the orbital integral, how to describe it as a geometric object. And as soon as we have done this, we will see that naturally the closed-index or minimized-index will come in the game even before doing any analysis. So given a semi-simple element gamma in the group G, I introduce the displacement function d of X and gamma X. So d being the remanent distance on the symmetric space. So it is a fundamental fact, a consequence of negative curvature or non-positive curvature, that this function d of d, the distance in general, is convex on X cross X. Convex in this case means that when restricted to due this X, it's convex. And so in particular, the function d of X to gamma X, a distance of X to gamma X, is convex. And the fact that gamma is semi-simple just says that the minimizing set for this displacement function is a convex subset, non-empty convex subset of X. So this is just the same simplicity of gamma which tells you this. So actually in our case, X of gamma, the minimizing set for the displacement function is actually the symmetric space for the centralizer z of gamma. So in other words, if you take a semi-simple element gamma and you look at the symmetric space associated with centralizer, the symmetric space itself is the total idio-disiq sum manifold of the manifold X. So how can we understand geometrically what is the orbital integral? So I wrote here X of gamma. So I take a judizik at X0 which is fixed, which is normal to X of gamma. And I take its image by gamma. So X of gamma is preserved by gamma, so this image is another judizik starting at gamma X0. Now because of negative curvature, these two judiziks are going to go far away. That is, the mutual distance of Y and gamma Y grows at least like the norm of Y, like the distance to X of gamma. So what is the orbital integral? The orbital integral is just the integral essentially of the heat color of X. Evaluate on Y and gamma Y, and you integrate just in all the possible normal directions with R of Y here being coefficient, which is a Jacobian. That's a geometric interpretation of the orbital integral. The fact that the orbital integral exists just uses the fact that the heat kernel at large distance decays like a Gaussian and compensates the possible growths of the decoding. So the proof we will give for the evaluation of this orbital integral will consist in pushing properly the integral to X of gamma by a deformation process. So let me now, well maybe I skip this, and let me give now a second fundamental identity, which is the fact that in effect the original orbital integral for the heat kernel, the elliptic one, the classical one, is given by the new orbital integral for the high-politic heat kernel. You can think of this as being an orbital integral for a nifidic dimensional vector bundle on the raise X. So these two orbital integrals are the same. And now we will make B tend to infinity in order to localize. It will localize to X gamma. So let me now explain the limit as B tend to infinity. So when you look at this high-politic Laplacian, you find that the dominating term as B tend to infinity is given by the jubizic flow after rescaling, so which forces in some sense the dynamics of the heat kernel to concentrate on closed-edizics. So let me now explain the final formula. So to explain the final formula, you can entirely forget about everything I said before. So let me take gamma to be a semi-sepal element, which I write in a canonical form after conjugation as a product of e to the a times k minus 1, where a is in B, k is in the group k, and a and k commute. Let me introduce z of gamma to be the centralizer of gamma, whose le algebra splits over the carton splitting as p of gamma plus k of gamma. So what is the formula for the semi-sepal orbital integrals, at least for the heat kernel? So the formula says the following. There is an explicit function, j gamma, which is evaluated on the k part of the le algebra, the centralizer of gamma. So you take this element gamma, you take this le algebra, you split it with p part and k part, and the orbital integral will be evaluated as an integral on the k part of the le algebra. So this is the orbital integral for the heat kernel. So this is an integral which comes from the geometry of x. And the right hand side is... Okay, so this exponential minus a square over 2t, this corresponds in some sense to the lengths of the due to the length squared associated with gamma, but above all what we have is an integral of k of gamma. And this integral of k of gamma, well there is a heat kernel here, it's not too bad. Here we have... Okay, I changed slightly gears because I introduced an homogeneous representation of k, and now I view my chasmir as acting on the sections of an extra vector volume. So this I did not tell you before, I was working before just with a lot of function functions. I also introduced now twisting, I just twist everything by representation of k. So we have a certain term which looks like a chasm character, like an equivalent chasm character. And so what when k easily believes that this is like a chasm character, this function j gamma will be like a thought form, or like an a-roof form. So note again the integral k gamma part, which is something which is in some sense invisible to classical analysis, it's somewhat ironic, that ultimately the final answer is expressed in terms of integral of k of gamma. So this formula accounts for the first term in Zellbrich's Trace formula I gave in the case of the main surfaces. So what is a function j gamma? It's a function again of the k part of the le algebra of the centralize of gamma. Now it looks very impressive, it is not. Well, I mean, I told you that we have this vector bound on tx plus m, sorry tx plus m. I would like to say that we have actually tx minus m. So this term j of gamma, it's a function of course, but it should be thought of as a ratio of the a-roof for tx divided by the a-roof of m. So it's really not like the classical Lefcet's formula, this tx plus m smithing propagates until the very end, and so we have the quotient here. So to make things more precise, in the case where gamma is trivial, if gamma is equal to 1, then the function j gamma of y0 is equal to the a-roof of i-ad of y0 acting on p divided by a-roof of i-ad of y0 acting on k. So in other words, instead like in classical Lefcet's formulas where we integrate differential forms, what we now integrate is functions under the algebra, and we integrate them whole, I mean, on the full k of gamma. So again the obvious analogy with a t-adbott, I mean, the formula, a fixed point formula of a t-adbott, is a pairing of a-roof of tx with a convention character of e. What we have here is just a sort of fake a t-adbott formula where of course the content is entirely different. And so tx in particular is replaced by tx minus m. Now, the last part is the connection with the loose space of x. So let x to be a remainder manifold and lx to be its loose space. So the loose space will be just loose maps from s1 to x at that stage. So as we know, lx is a remainder manifold, we can equip it naturally with the l2 metric, and s1 acts isometrically on lx, and the generator of the action is k of x, which is x dot. So in particular the zero set of this generator, and the trivial loops, they are point x, that's x, so x sits in lx as a zero set of a vector field, of a killing vector field, and a normal bundle to x in lx are just functions. You just take a point x, you take the tangent bundle, txx, and you look at all functions from s1 into this fiber whose integral is equal to zero. That's a normal bundle to x to lx. And now what I shall do is evaluate the inverse of the equivalent overclass of the normal bundle to lx in lx. Well, I mean, we are in the context where you could apply equivalent chronology, all these vector bundles are acted upon by s1. So the only difficulty is that since this normal bundle is infinite dimensional, we would have an infinite product. But if the infinite product is suitably normalized, this was a remark of Witton and Thierre that you get exactly the A-roof genius of tx. So in other words, the inverse of the equivalent overclass of the normal bundle is just given by a known characteristic class. So index theory and localization formulas. So I'm going back here to a Thierre Witton and later work I did on this, which is to say that you can interpret the heat equation formula. When you look at the heat equation formula for the index, it just looks like a magic analytic trick. Now, if you view this from a physicist's point of view, you can rewrite this formula at the first stage in a rigorous way as an integral of some measure. But actually, if you do the physicist's way, you just rewrite it as an integral of an explicit differential form. This differential form does not exist formally, but still it has a proper symmetry. It is essentially closed. It is equivariately closed under the action of s1. This was the observation of a Thierre that if you apply localization formulas in equivine commonality, what you get is a true answer, which is to say that the index of the x is equal to the integral of x of A-roof stancher. So it's as if the mechanism of equivine commonality was already at work in implicit form in index theory. So the form beta is here, the alpha t, maybe I should just explain what it is briefly. It is an explicit form which exists universally on any space with an action of s1. It contains as a leading term here essentially alpha t. It contains as a leading term exponential minus k squared over t, e to t. There is an extra term of degree 2. Bj is also explicit. It is a lift as an equivine form of the turn character form of e. And so if you apply formally without doing a limit, localization formulas in equivine commonality, you get the index formula. So at the time I was upset with this sort of argument, but later on I convinced myself that what the heat equation is doing to prove the index theorem is just doing a loop space, what you would do on any space. So in other words, it's not a loop that the analysis has invented some new procedure. It is the opposite which is true. It is just exploiting the best of its capabilities what exists universally on any space equipped with an action of a lead group. Now, let me go to orbital theory of the loop group. So let G be a compacted connected loop group, sorry, group, compacted connected lead group, not loop group. And let Lg, sorry, sorry, Lg, is its loop group that just moves maps from S1 into G. So I will not here introduce a central extension. I prefer to introduce the, for the moment, L till the G which is a semi-direct product of Lg with S1. We're taking into account that S1 acts naturally on Lg. So the le algebra of this extended lead group is now an algebra of differential operators on S1. So it is spanned linearly, the first piece which is d over dt, and the second piece which is at, where at is a periodic section of the le algebra. So actually, essentially the coidroid orbits of the extended loop group to respond to connections. So this I will explain with a homonymy lies in an orbit in the group. Well, because if you look at d over dt plus at, this is exactly a connection. So you can view in some sense the elements of the le algebra as being connections of the trivial G bundle. And the coidroid orbit corresponds to connections of the trivial G bundle, whose an anonymy lie in a conjugacy class in the orbit G. So if instead of looking at the connections, you look at its homonymy, you just integrate the, by the parallel transport for the connection. So in other words, you integrate the equation d over dt plus at of gt equal to 0, which is 0 equals 1. What you get is a pass which connects in the group, the identity to an orbit in the group. So in some sense, the coidroid orbits of the loop group are just the passes which connect the identity to an orbit in the group. So these are naturally symplatic metaphors. So heat current localization. So if I might compactly group by introducing a very metric, delta G is a fraction, and I can see that heat current on the loop group. So pt of g, just if you write it the way that physicists do, and I'm sure the way that you would do it, you get the pt of g is an integral over this orbit space of exponential minus d energy over t times dg. So I claim that this integral can be transformed into a Bellin-Vell integral. Because indeed, the energy of a pass corresponds just to the Hamiltonian for the natural action of s1 on the coidroid orbit. And so of course we don't know what dg is, but for lack of a better measure, we will pretend that dg is just a symplatic measure on the orbit. So if you apply again formally the localization formula as the equivalent of homology, just at the formal level, what you get is a formula which expresses pt of g as an integral over the fixed point sets of this action of s1 here, of the action of rotation, and that are just one parameter semi-groups. So actually you obtain again a known formula, you obtain a formula which is correct, which seems to say that the equivalent of homology again applies in this context. So let me give a remark by Igor Frankl and Atiyah. So Igor Frankl had observed that the heat kernel appears in the numerator of the left sets formula for the, well I should say the central extension of lg. So when you compute the character formulas, essentially what you have in the numerator is a heat kernel, and he suggested that this reflects a Kirinoff to left sets fixed point principle. Well pt of g again formally is an integral group space, and left sets formulas for character groups is just a sum over the affine line group. And if you look at this, in fact that these two things are equal, you see that at least formally this reflects a Kirinoff left set fixed point principle. So Atiyah suggested that the localization formula should apply as to the integral, and so what does the formal proof of localization will consist of? So how will we deal rigorously with this question of knowing does localization formula techniques apply to these integrals over the loop group? So if you look at what k is in this case, so the points of the point order of it are just connections, v over dt plus at, the vector field which consists in looking at the shift of the connection is just a dot, so that k squared is equal to the integral of a dot square, and so ultimately knowing that a is equal to g dot, what you see appearing is integral of g double dot square. So actually if you just formally try to report the message known to put localization formulas in equivocal ecology, you're unavoidably glad to look at such a pathogen rule which is a regularised pathogen rule, and which corresponds in the proper way to a hyperlinked So this I would expect. So the formal consequences, introducing the costant Dirac operator is a way to prove the Fourier formula for PTRG using localization formulas. In the context of symmetric space of non-compact time, the same mechanism can still be used but it's subtler. In other words, the whole question of descent from a reductive group to its compIc quotient to its quotient by k is not incorporated in the traditional mechanism of localization formulas. Okay, so here are just a few references, and that's it. Thank you very much. Thank you. Well, I mean, you know, if you have taken down the question of taking non-semi-semi-semi elements, of course, I mean, it's fundamental. But in some sense, that's a different question. That's a different question. That is the proper way to approach it is in some sense to approximate a non-semi-semi element by some simple element, truncate the interval, and then see what happens. This method will not, it will not delete the fundamental analytic and geometric difficulties they are when you go to the cusp, and in particular to iterated cusps where the structure of the boundary becomes very complicated. So this you will still have to work. You will discover them in a different language, and they will still be there. Thank you very much.
 It's going to be difficult.  Maybe I'll redo it.  OK, so on this left-hand side, you should  think that there is an integral over the loop space.  And on the right-hand side, just a sum over selected loops,  which are precisely the closed-ruthy 6.  And it is in this sense that eventually the interpretation  of the proof and of the formula I will give  will be sort of as a localization formula  in equilibrium homology, where on the left-hand side,  we have something evaluated on the full loop space,  integral full loop space.  And on the right-hand side, we evaluate  simply sums over closed-ruthy 6.  So that's a stupid question.  But when you say equal, these things exactly equal.  Exactly equal.  And that's a funny expansion.  No, no, it's not.  It's not a funny expansion.  In this case, it's exactly equal.  It's as equal as it can be.  It's an equality that we will eventually  describe as an identity of the same sort  as an index formula in some way.  So that's an equality.  So the purpose of the talk will be  to extend the very explicit formula  to a between locally symmetric spaces,  explain how to obtain the formula by an interpolation  process.  So we will actually interpolate smoothly  between the left-hand side and the right-hand side.  That connects with infinite dimensional version  of Deuysse Weifengmann-Berlin-Wang formulas.  So first of all, left-sheds formulas.  So I will start with some very general arguments.  So Euler characteristics.  So I take x to be a compact manifold.  And I define the Euler characteristic.  So as you know, there are many ways  of computing the Euler characteristic,  either through a triangulation, but there  is another way through the heat kernel.  So you introduce basically the Hodge-Loplaschen square x  associated with a given remanian metric on the manifold.  And we have this basic formula, which  says that the Euler characteristic of x  can be evaluated in terms of traces of certain heat kernels.  Actually, there is a sign TRS, which means supertrace.  It's graded trace.  It's alternating sum of traces.  And this is for the heat kernel of the Hodge-Loplaschen  acting on the various forms of various degrees.  So you take the alternating sum of these traces,  and you get the Euler characteristic.  This formula is quite easy to prove.  So actually, one of the goals, one  of the uses of such a formula, in this case,  can sit in making s 10 to 0.  And this way, you will obtain Schoen-Gausbohne.  So more or less, this is a scheme of how  to obtain the other heat equation all index formula.  Now, left sets formula.  So I take slightly more refined thing.  I say g to be a diffeomorphism in diffeomorphism of x.  And so I know that g acts on the quorumology of the manifold.  So then you can define the left sets number.  That's the alternating sum of the traces  of the action of g on the quorumology.  This is a global invariant.  So again, you still have a formula of the type of McKinsey,  which says that the left set number, which  is something which only involves the quorumology,  can be evaluated in much, with a much bigger object, which  is the full-term uncompact, just as before.  And the formula is almost the same,  except that you introduce here the action of the diffeomorphism  in the trace.  And again, such formulas are quite easy to prove.  So by making s then to 0, we obtain l of g  as a sum of local sum of contributions  of the fixed point of g.  That's the left sets formula.  So I wrote here an interpolation formula,  in which here there is a g missing.  So the interpolation here is meant  to do the following thing.  If you make s then to infinity here,  you obviously get the left set number,  because you obviously project on the harmonic forms, which  represent the quorumology.  And on the right-hand side, we make the limit as s then to 0.  You obtain whatever formula there is for the left set number  of g.  So at the same time, this object remains constant  and interpolates naturally between the left-hand side  and the right-hand side.  So this is exactly as the mental mechanism  will apply in the case of the trace formula.  So let me now take a compact manifold, x compact remanian  manifold.  Let me introduce delta x to be the Laplace-Baltrami operator  on x.  And then for fixed t, I just call g the heat current.  So I deliberately use a segmentation  that I used before for the nithiomorphism.  So I'm going to ask four questions.  So the first question is, is the trace of the heat kernel  an Euler characteristic?  What I mean by this is, can I think of the trace  of the heat kernel as being an Euler characteristic of exactly  the same type as before, just by doing the proper changes?  Now, the second question is, can I express this Euler  characteristic, or if you like, is generalizing the heat  current using a much bigger object?  So before, in the case of the chronology,  we have the chronology, and we resolved it  by the Durand complex.  I'm asking, can I resolve in a certain way the space of smooth  functions with values in R by a certain complex  and re-express the trace here in terms of alternating some  of traces on a much larger object?  So the analogy again is passing from the chronology  to the Durand complex.  We want to pass here from the smooth functions  to some complex which will resolve the smooth functions  on X. And the third question will be,  by making b tend to infinity, do we obtain  Zellberg's trace formula?  So in other words, can we think of the same type of trace  formula as a generalized index formula?  So again, I gave here the interpolation scheme.  Here is a trace here on smooth functions  of g of the heat kernel, which is a global invariant.  By making b tend to infinity, we still  don't know what the parameter is.  We will obtain Zellberg's trace formula  that we will think of as being a local formula.  So what this means by analogy is that we will have to think  of the smooth functions as the homology of some complex  and even more as the harmonic objects in some complex.  So I will try progressively to give the instruments which  allow us to do this.  So the answer is yes.  So first of all, the question of resolving the state  of smooth functions.  So in the case of a manifold, again, we took the homology  and we resolved it in the proper sense  by the Durand complex.  Here we're going to ask, can we think of the smooth functions  on x as being the homology of something?  So let me construct an infinite number  of ways of doing this.  So I just give myself a real vector bundle on x.  And I call this script to be the total space of E.  And then I introduce the space R. So this  is the relative Durand complex.  That the smooth forms are the total space,  but which are just forms along the fiber.  So this family of Durand complexes along the fiber.  And DE is just a fiber-wise Durand complex.  Now, I claim that my plurgery is lemma.  The co-homology of this complex is precisely  the smooth functions on the base.  Well, this is because the co-homology of the Durand  complex itself is made.  Ultimately, it comes at a degree 0.  It is a function which are constants, fiber-wise,  meaning that they come from the base.  So this is a complex whose co-homology is the smooth functions.  Now, does this complex have a Hodge theory?  Does this new complex R?  Does it have a Hodge theory?  So I claim that there is such a Hodge theory,  because if you introduce GE to be an Euclidean metric,  there is a Hodge theory for all these vector fibers.  And this Hodge theory is in some sense  given by the Wittner-Lapplation.  So just a short reminder about Wittner-Lapplation.  So in very simple case, I give myself  E to be an Euclidean vector space.  The classical obvious Hodge-Lapplation  has continuous spectrum.  But in some sense, the Wittner-Lapplation  has discrete spectrum.  So you can view it in one way, is to twist the Durand operator  by the Gaussian function.  Do not change the co-homology this way.  So you construct the corresponding Hodge-Lapplation  by taking the formula joint.  What you get, essentially, is a Laplacian.  You get a first operator, which is H,  which is a harmonic oscillator, and N, which  counts the degree in the exterior algebra.  And so you recover, in some sense,  the proper computation of the co-homology of a vector space,  which is concentrated in degree 0, because H is the kernel  of H is one dimensional, and N, 0 forms.  So just to remark here that via the so-called Bergman's  isomorphism, you can instead replace the Durand complex  of E by the algebraic Durand complex on E.  That is, this complex is quasi-isomorphic to this algebraic  Durand complex.  And then we see appearing here the polynomials, which  are the symmetric algebra.  And as we know, which is a natural inverse  to the exterior algebra.  So this construction here will play an important role  in the cycle.  So now we have our base manifold, X,  and the fiber bundle, which is E. And the question  will be to be how to couple the base and the fiber  so that we can do something.  So we need to introduce, first of all, the heat operator  on the base, X, coupled with the written Laplacian on E.  So this is the way that we'll approach  the construction of this ultimately  high-polyptical Laplacian, how to couple the base, which  contains the heat kernel, with the fiber, which  contains the written Laplacian.  So let me explain the construction explicit in the case  of a symmetric space.  So let G be a real reductive group,  K to be a maximal compact subgroup, and X  equal G over K to be the symmetric space.  I then introduce a corresponding carton splitting  equipped with the non-generated bill in a form B.  And this carton splitting descends to a vector bundle, E,  which is made of two pieces, TX plus M. So it's a general fact  that on a symmetric space or locally symmetric space,  the tangent bundle has a natural auto complement, which  is this thing which I call N. TX is modeled on B,  and N is modeled on K. And actually,  our interpolation will ultimately  be built on the total space of such a vector space,  of such a vector bundle.  So we will explain how to couple X and TX plus M  by lifting first everything to G multiplied by the realtor.  So just an example in the case where G is a sum of R, K is S1,  X is the hyper-alf-plane, and TX plus M is of dimension 3,  so that the total dimension of fiber bundle  is of dimension 5.  So how are we going to construct this coupling?  So Casimir and Kostalt.  So first of all, on the reductive group,  we have the Casimir operator, which  is a second-order differential operator acting on the group,  which is neither elliptic, neither hyperbolic.  So it is, in some sense, positive in the p-direction  and negative in the  k-direction.  That's a second-order differential operator,  and which is in the center of the vector space.  It is in the center of the enveloping algebra.  Now I introduce the clefted algebra C hat of G,  which is the clefted algebra of G minus B.  And as we know, this clefted algebra acts on the  exterior algebra of G stuff.  So I also introduce U of G, which is the enveloping algebra.  So if you like it, the algebra of differential operators  of invariant differential operators of the group G,  and the clefted Kostalt-deroc operator is a section  of the clefted algebra tensed with the enveloping algebra.  So it's a slight but significant modification  of the construction of the classical deroc operator.  It is constructed via the fundamental closed  three-form of the group G, which I wrote here.  And here is a short formula for it.  You don't need to know this, so this is just, in some sense,  the classical deroc operator, where the EIs are just  differential operators acting on G.  And here there is a cubic correction in terms of the  clefted image of the three-form.  Now the basic result of Kostalt is that the square  of the Kostalt operator, with my conventions,  is equal to minus the casimir.  You could have another convention in which you would  get the plus the casimir, plus some irrelevant constant.  So what this means is that the casimir operator  has a natural differential operator square root.  And this we will exploit in the context of analysis.  So let me observe that the Kostalt operator,  what it acts upon, it acts on smooth functions on G  with values in the exterior algebra of G star,  while the casimir operator itself acts on the smooth functions  on G with values in R. So in some sense we get a sort of  straight root of the Laplacian, but it acts on the wrong space.  So it's like straight root of minus one, which was just  outside the real line.  Kostalt, because I had K for, okay, I thought of this.  I had K to be the maximal compact subgroup,  and I did not want to have the question,  what is K Kostalt?  So this was made for you.  Okay, so the operator Db now.  So let me construct a sort of generalized Dirac operator.  This Dirac operator will act on G product-ally algebra.  So this new Dirac operator will act on smooth functions  on G tensor, the types of the algebra,  with values in lambda of G star.  Okay, so, sorry, this is going too fast.  So here is the formula.  So I have a first term, which is just a Kostalt Dirac operator.  So this one acts on  the infinity of G tensor with lambda of G star.  This is for the first piece.  This one, you should not look at it.  Absolutely not.  Do as if it was not there.  So this piece here is just a version of the Hodg-Durham  written operator.  That is, that's a sort of D plus D star, if you like,  but in the proper Hodg theory.  That is, in the proper Hodg theory of P  and in the proper Hodg theory of K,  which involves a written twist.  Now you see that these two pieces, P and K,  are treated differently.  I mean, this different treatment reflects the fact  that we are on a reductive group.  But essentially, the effect when you square this  is just to produce a very good Hodg-Lop-Lasch.  So in some sense, these D papers, D per star,  it acts on sin infinity of G of P, sorry,  with coefficients in lambda of P star.  And the other one acts on sin infinity of K  with coefficients in lambda of K star.  So if you combine all of these, you find that this new  operator, D B, which I will call, for lack of a better name,  a Dirac operator, acts on the smooth function of G cross G  with values in lambda of G star.  So what is the high-polyptical operation in this context?  So what I'll do for the moment, I'm working on G,  on the group G, 10 times the l-algebra G.  So I take the square of the previous operator,  D B square, and then I subtract from this square,  the square of cos dot.  So if you like, I do A plus B square minus A square.  And here is a formula for this operator that I will explain,  I will show to you.  But before doing this, I descent, I quotient,  the whole construction by K.  So in other words, for the moment, we've done analysis  on the group times the l-algebra, but then we have to  descend everything to the symmetric space.  So as I explained before, the l-algebra G descends to Tx plus  m, G with its cut-out splitting, and G cross the l-algebra  to the total space x hat of Tx plus m.  So our new operator, lbx, which is obtained by descent,  will act on the smooth functions on x hat,  with values in the exterior algebra of T star x plus m star.  So again, using the Fibromois-Bartmann isomorphism,  instead of viewing the construction as acting on a bigger space,  which is a total space of a vector bubble,  you can instead view this operator as acting on an  infinite-dimensional vector bubble.  The infinite-dimensional vector bubble is just the symmetric  algebra of T star x plus m star, multiplied by the exterior  algebra of T star x plus m star.  So here is the formula for this lbx.  So this lbx contains along the fiber the harmonic oscillator  of Tx plus m, but in some sense along the fiber it is just  the written population.  What we see appearing here as an operator which just  differentiates in the base direction is actually the judizic flaw.  And we have a bunch of other matrix terms that I will not explain.  There is also this extra term here, which is a term of order four,  which means that our operator, we are quitting the  unshattered word of harmonic oscillators to go to something  slightly more complicated of operators with coefficients  quadratic, but this piece is absolutely necessary to make the theory work.  So this operator is hyperliptic.  So what does this mean hyperliptic here?  That means that it does a good analysis in spite of the fact  that it just differentiates twice in the fiber direction,  but only differentiates once in the base direction.  So analytically it is good, it is part of the class of operators  concerned, considered by Hermandem.  And so what I'm saying, and I will not show this,  is that in the proper sense this operator deforms  the original Casimir operator acting on X,  or if you like deforms the original Laplacian.  There is a collapsing phenomenon which is relatively subtle,  which means that analytically splitting this operator deforms  the original Laplacian.  But now the wonderful thing is that when B tends to infinity,  it will be the judizic flaw which will dominate.  And so when B tends to infinity, this will force the localization  on closed physics when you consider traces of it cones.  So the case of locally symmetric spaces now,  so let me take gamma to be a compact torsion free  discrete subgroup of G, and then we consider the quotient  of X by gamma, which is a compact locally symmetric space.  So all the constructions I have done, so I have done constructions  over G, then I descended these constructions over X,  and now of course I can descend the construction to the quotient,  because basically all the objects I introduced are properly equivariable.  And so the fundamental identity is to say that the trace,  in some sense of ordinary heat kernel, so Cz,  in some sense you can think of this as a heat kernel,  now Cz minus C, as being the, this is a sort of,  I mean it's just modification of heat kernel by constant,  is equal to the super trace of the corresponding  hyperliptic operator.  So exactly like an index theory where you say the original index,  or Fshad number, is equal to the super trace of a heat operator  acting on a bigger space, we have exactly the same sort of phenomenon here,  and the proof is in this context rather elementary,  it relies on the fact, on the collapsing of this operator  to this one as B10 to zero, plus the invariance of the super trace  under deformation by B, which is essentially an index theoretic proof.  However this is not good enough for our purpose,  because in some sense we're not going to apply to play the Zellberg's trace formula  in its algebraic version.  So actually when you have the trace of each kernel,  locally symmetric space, you know that you can re-express it  as a sum of objects which are evaluated on the symmetric space.  So you can go back to the symmetric space.  So in other words, in this equality, we can as well evaluate  the left hand side and the right hand side as being sums,  infinite sums, over the district root gamma.  And now the true miracle is, is that it's not only true  that the trace on the compact locally symmetric space  is equal to the super trace again over the compact locally symmetric space,  but orbital integral by orbital integral, there is a preservation.  So now we're going to move to different sort of geometry.  So the identity splits as an identity over orbital integrals,  and this is why we will be able eventually to give geometric formula for that.  So semi-simple orbital integrals.  Let me take gamma to be semi-simple in G,  and let me call gamma, right, it's country AC class in G.  Then you can, this has already been introduced here,  introduced in the proper sense, the orbital integral,  in this case of the heat kernel, which is an integral  of the quotient of G by the centralizer of gamma in the group.  So in the proper sense, this integral makes sense,  in spite of the fact that Pt of X is just a heat kernel of X  over the symmetric space.  So what I'm going to try to explain is how first of all to geometries  the orbital integral, how to describe it as a geometric object.  And as soon as we have done this, we will see that naturally  the closed-index or minimized-index will come in the game  even before doing any analysis.  So given a semi-simple element gamma in the group G,  I introduce the displacement function d of X and gamma X.  So d being the remanent distance on the symmetric space.  So it is a fundamental fact, a consequence of negative curvature  or non-positive curvature, that this function d of d,  the distance in general, is convex on X cross X.  Convex in this case means that when restricted to due this X,  it's convex.  And so in particular, the function d of X to gamma X,  a distance of X to gamma X, is convex.  And the fact that gamma is semi-simple just says that the minimizing set  for this displacement function is a convex subset,  non-empty convex subset of X.  So this is just the same simplicity of gamma which tells you this.  So actually in our case, X of gamma,  the minimizing set for the displacement function is actually  the symmetric space for the centralizer z of gamma.  So in other words, if you take a semi-simple element gamma  and you look at the symmetric space associated with centralizer,  the symmetric space itself is the total idio-disiq sum manifold of the manifold X.  So how can we understand geometrically  what is the orbital integral?  So I wrote here X of gamma.  So I take a judizik at X0 which is fixed, which is normal to X of gamma.  And I take its image by gamma.  So X of gamma is preserved by gamma,  so this image is another judizik starting at gamma X0.  Now because of negative curvature,  these two judiziks are going to go far away.  That is, the mutual distance of Y and gamma Y grows at least  like the norm of Y, like the distance to X of gamma.  So what is the orbital integral?  The orbital integral is just the integral essentially of the heat color of X.  Evaluate on Y and gamma Y,  and you integrate just in all the possible normal directions  with R of Y here being coefficient, which is a Jacobian.  That's a geometric interpretation of the orbital integral.  The fact that the orbital integral exists just uses the fact that the heat kernel  at large distance decays like a Gaussian  and compensates the possible growths of the decoding.  So the proof we will give for the evaluation of this orbital integral  will consist in pushing properly the integral to X of gamma by a deformation process.  So let me now, well maybe I skip this,  and let me give now a second fundamental identity,  which is the fact that in effect the original orbital integral for the heat kernel,  the elliptic one, the classical one,  is given by the new orbital integral for the high-politic heat kernel.  You can think of this as being an orbital integral  for a nifidic dimensional vector bundle on the raise X.  So these two orbital integrals are the same.  And now we will make B tend to infinity in order to localize.  It will localize to X gamma.  So let me now explain the limit as B tend to infinity.  So when you look at this high-politic Laplacian,  you find that the dominating term as B tend to infinity is given by the jubizic flow  after rescaling, so which forces in some sense the dynamics  of the heat kernel to concentrate on closed-edizics.  So let me now explain the final formula.  So to explain the final formula, you can entirely forget about everything I said before.  So let me take gamma to be a semi-sepal element,  which I write in a canonical form after conjugation  as a product of e to the a times k minus 1,  where a is in B, k is in the group k,  and a and k commute.  Let me introduce z of gamma to be the centralizer of gamma,  whose le algebra splits over the carton splitting as p of gamma plus k of gamma.  So what is the formula for the semi-sepal orbital integrals,  at least for the heat kernel?  So the formula says the following.  There is an explicit function, j gamma,  which is evaluated on the k part of the le algebra, the centralizer of gamma.  So you take this element gamma, you take this le algebra,  you split it with p part and k part,  and the orbital integral will be evaluated as an integral on the k part of the le algebra.  So this is the orbital integral for the heat kernel.  So this is an integral which comes from the geometry of x.  And the right hand side is...  Okay, so this exponential minus a square over 2t,  this corresponds in some sense to the lengths of the due to the length squared associated with gamma,  but above all what we have is an integral of k of gamma.  And this integral of k of gamma, well there is a heat kernel here,  it's not too bad.  Here we have...  Okay, I changed slightly gears because I introduced an homogeneous representation of k,  and now I view my chasmir as acting on the sections of an extra vector volume.  So this I did not tell you before, I was working before just with a lot of function functions.  I also introduced now twisting, I just twist everything by representation of k.  So we have a certain term which looks like a chasm character, like an equivalent chasm character.  And so what when k easily believes that this is like a chasm character,  this function j gamma will be like a thought form, or like an a-roof form.  So note again the integral k gamma part,  which is something which is in some sense invisible to classical analysis, it's somewhat ironic,  that ultimately the final answer is expressed in terms of integral of k of gamma.  So this formula accounts for the first term in Zellbrich's Trace formula I gave in the case of the main surfaces.  So what is a function j gamma?  It's a function again of the k part of the le algebra of the centralize of gamma.  Now it looks very impressive, it is not.  Well, I mean, I told you that we have this vector bound on tx plus m, sorry tx plus m.  I would like to say that we have actually tx minus m.  So this term j of gamma, it's a function of course,  but it should be thought of as a ratio of the a-roof for tx divided by the a-roof of m.  So it's really not like the classical Lefcet's formula,  this tx plus m smithing propagates until the very end, and so we have the quotient here.  So to make things more precise, in the case where gamma is trivial,  if gamma is equal to 1, then the function j gamma of y0 is equal to the a-roof of i-ad of y0 acting on p  divided by a-roof of i-ad of y0 acting on k.  So in other words, instead like in classical Lefcet's formulas where we integrate differential forms,  what we now integrate is functions under the algebra, and we integrate them whole, I mean, on the full k of gamma.  So again the obvious analogy with a t-adbott, I mean, the formula, a fixed point formula of a t-adbott,  is a pairing of a-roof of tx with a convention character of e.  What we have here is just a sort of fake a t-adbott formula where of course the content is entirely different.  And so tx in particular is replaced by tx minus m.  Now, the last part is the connection with the loose space of x.  So let x to be a remainder manifold and lx to be its loose space.  So the loose space will be just loose maps from s1 to x at that stage.  So as we know, lx is a remainder manifold, we can equip it naturally with the l2 metric,  and s1 acts isometrically on lx, and the generator of the action is k of x, which is x dot.  So in particular the zero set of this generator, and the trivial loops, they are point x,  that's x, so x sits in lx as a zero set of a vector field, of a killing vector field,  and a normal bundle to x in lx are just functions.  You just take a point x, you take the tangent bundle, txx,  and you look at all functions from s1 into this fiber whose integral is equal to zero.  That's a normal bundle to x to lx.  And now what I shall do is evaluate the inverse of the equivalent overclass  of the normal bundle to lx in lx.  Well, I mean, we are in the context where you could apply equivalent chronology,  all these vector bundles are acted upon by s1.  So the only difficulty is that since this normal bundle is infinite dimensional,  we would have an infinite product.  But if the infinite product is suitably normalized, this was a remark of Witton and Thierre  that you get exactly the A-roof genius of tx.  So in other words, the inverse of the equivalent overclass of the normal bundle  is just given by a known characteristic class.  So index theory and localization formulas.  So I'm going back here to a Thierre Witton and later work I did on this,  which is to say that you can interpret the heat equation formula.  When you look at the heat equation formula for the index, it just looks like a magic analytic trick.  Now, if you view this from a physicist's point of view,  you can rewrite this formula at the first stage in a rigorous way as an integral of some measure.  But actually, if you do the physicist's way,  you just rewrite it as an integral of an explicit differential form.  This differential form does not exist formally,  but still it has a proper symmetry.  It is essentially closed.  It is equivariately closed under the action of s1.  This was the observation of a Thierre that if you apply localization formulas in equivine commonality,  what you get is a true answer, which is to say that the index of the x is equal to the integral of x of A-roof stancher.  So it's as if the mechanism of equivine commonality was already at work in implicit form in index theory.  So the form beta is here, the alpha t, maybe I should just explain what it is briefly.  It is an explicit form which exists universally on any space with an action of s1.  It contains as a leading term here essentially alpha t.  It contains as a leading term exponential minus k squared over t, e to t.  There is an extra term of degree 2.  Bj is also explicit. It is a lift as an equivine form of the turn character form of e.  And so if you apply formally without doing a limit, localization formulas in equivine commonality, you get the index formula.  So at the time I was upset with this sort of argument, but later on I convinced myself that what the heat equation is doing  to prove the index theorem is just doing a loop space, what you would do on any space.  So in other words, it's not a loop that the analysis has invented some new procedure.  It is the opposite which is true. It is just exploiting the best of its capabilities what exists universally on any space  equipped with an action of a lead group.  Now, let me go to orbital theory of the loop group.  So let G be a compacted connected loop group, sorry, group, compacted connected lead group, not loop group.  And let Lg, sorry, sorry, Lg, is its loop group that just moves maps from S1 into G.  So I will not here introduce a central extension.  I prefer to introduce the, for the moment, L till the G which is a semi-direct product of Lg with S1.  We're taking into account that S1 acts naturally on Lg.  So the le algebra of this extended lead group is now an algebra of differential operators on S1.  So it is spanned linearly, the first piece which is d over dt, and the second piece which is at,  where at is a periodic section of the le algebra.  So actually, essentially the coidroid orbits of the extended loop group to respond to connections.  So this I will explain with a homonymy lies in an orbit in the group.  Well, because if you look at d over dt plus at, this is exactly a connection.  So you can view in some sense the elements of the le algebra as being connections of the trivial G bundle.  And the coidroid orbit corresponds to connections of the trivial G bundle,  whose an anonymy lie in a conjugacy class in the orbit G.  So if instead of looking at the connections, you look at its homonymy,  you just integrate the, by the parallel transport for the connection.  So in other words, you integrate the equation d over dt plus at of gt equal to 0, which is 0 equals 1.  What you get is a pass which connects in the group, the identity to an orbit in the group.  So in some sense, the coidroid orbits of the loop group are just the passes which connect the identity to an orbit in the group.  So these are naturally symplatic metaphors.  So heat current localization.  So if I might compactly group by introducing a very metric, delta G is a fraction,  and I can see that heat current on the loop group.  So pt of g, just if you write it the way that physicists do, and I'm sure the way that you would do it,  you get the pt of g is an integral over this orbit space of exponential minus d energy over t times dg.  So I claim that this integral can be transformed into a Bellin-Vell integral.  Because indeed, the energy of a pass corresponds just to the Hamiltonian for the natural action of s1 on the coidroid orbit.  And so of course we don't know what dg is, but for lack of a better measure,  we will pretend that dg is just a symplatic measure on the orbit.  So if you apply again formally the localization formula as the equivalent of homology,  just at the formal level, what you get is a formula which expresses pt of g as an integral  over the fixed point sets of this action of s1 here, of the action of rotation,  and that are just one parameter semi-groups.  So actually you obtain again a known formula, you obtain a formula which is correct,  which seems to say that the equivalent of homology again applies in this context.  So let me give a remark by Igor Frankl and Atiyah.  So Igor Frankl had observed that the heat kernel appears in the numerator of the left sets formula  for the, well I should say the central extension of lg.  So when you compute the character formulas, essentially what you have in the numerator is a heat kernel,  and he suggested that this reflects a Kirinoff to left sets fixed point principle.  Well pt of g again formally is an integral group space,  and left sets formulas for character groups is just a sum over the affine line group.  And if you look at this, in fact that these two things are equal,  you see that at least formally this reflects a Kirinoff left set fixed point principle.  So Atiyah suggested that the localization formula should apply as to the integral,  and so what does the formal proof of localization will consist of?  So how will we deal rigorously with this question of knowing does localization formula techniques  apply to these integrals over the loop group?  So if you look at what k is in this case,  so the points of the point order of it are just connections,  v over dt plus at, the vector field which consists in looking at the shift of the connection is just a dot,  so that k squared is equal to the integral of a dot square,  and so ultimately knowing that a is equal to g dot, what you see appearing is integral of g double dot square.  So actually if you just formally try to report the message known to put localization formulas in equivocal ecology,  you're unavoidably glad to look at such a pathogen rule which is a regularised pathogen rule,  and which corresponds in the proper way to a hyperlinked  So this I would expect.  So the formal consequences, introducing the costant Dirac operator is a way to prove the Fourier formula  for PTRG using localization formulas.  In the context of symmetric space of non-compact time, the same mechanism can still be used but it's subtler.  In other words, the whole question of descent from a reductive group to its compIc quotient to its quotient by k  is not incorporated in the traditional mechanism of localization formulas.  Okay, so here are just a few references, and that's it. Thank you very much.  Thank you.  Well, I mean, you know, if you have taken down the question of taking non-semi-semi-semi elements, of course, I mean, it's fundamental.  But in some sense, that's a different question. That's a different question.  That is the proper way to approach it is in some sense to approximate a non-semi-semi element by some simple element,  truncate the interval, and then see what happens.  This method will not, it will not delete the fundamental analytic and geometric difficulties they are  when you go to the cusp, and in particular to iterated cusps where the structure of the boundary becomes very complicated.  So this you will still have to work. You will discover them in a different language, and they will still be there.  Thank you very much.
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It's going to be difficult. Maybe I'll redo it. OK, so on this left-hand side, you should think that there is an integral over the loop space. And on the right-hand side, just a sum over selected loops, which are precisely the closed-ruthy 6. And it is in this sense that eventually the interpretation of the proof and of the formula I will give will be sort of as a localization formula in equilibrium homology, where on the left-hand side, we have something evaluated on the full loop space, integral full loop space. And on the right-hand side, we evaluate simply sums over closed-ruthy 6. So that's a stupid question. But when you say equal, these things exactly equal. Exactly equal. And that's a funny expansion. No, no, it's not. It's not a funny expansion. In this case, it's exactly equal. It's as equal as it can be. It's an equality that we will eventually describe as an identity of the same sort as an index formula in some way. So that's an equality. So the purpose of the talk will be to extend the very explicit formula to a between locally symmetric spaces, explain how to obtain the formula by an interpolation process. So we will actually interpolate smoothly between the left-hand side and the right-hand side. That connects with infinite dimensional version of Deuysse Weifengmann-Berlin-Wang formulas. So first of all, left-sheds formulas. So I will start with some very general arguments. So Euler characteristics. So I take x to be a compact manifold. And I define the Euler characteristic. So as you know, there are many ways of computing the Euler characteristic, either through a triangulation, but there is another way through the heat kernel. So you introduce basically the Hodge-Loplaschen square x associated with a given remanian metric on the manifold. And we have this basic formula, which says that the Euler characteristic of x can be evaluated in terms of traces of certain heat kernels. Actually, there is a sign TRS, which means supertrace. It's graded trace. It's alternating sum of traces. And this is for the heat kernel of the Hodge-Loplaschen acting on the various forms of various degrees. So you take the alternating sum of these traces, and you get the Euler characteristic. This formula is quite easy to prove. So actually, one of the goals, one of the uses of such a formula, in this case, can sit in making s 10 to 0. And this way, you will obtain Schoen-Gausbohne. So more or less, this is a scheme of how to obtain the other heat equation all index formula. Now, left sets formula. So I take slightly more refined thing. I say g to be a diffeomorphism in diffeomorphism of x. And so I know that g acts on the quorumology of the manifold. So then you can define the left sets number. That's the alternating sum of the traces of the action of g on the quorumology. This is a global invariant. So again, you still have a formula of the type of McKinsey, which says that the left set number, which is something which only involves the quorumology, can be evaluated in much, with a much bigger object, which is the full-term uncompact, just as before. And the formula is almost the same, except that you introduce here the action of the diffeomorphism in the trace. And again, such formulas are quite easy to prove. So by making s then to 0, we obtain l of g as a sum of local sum of contributions of the fixed point of g. That's the left sets formula. So I wrote here an interpolation formula, in which here there is a g missing. So the interpolation here is meant to do the following thing. If you make s then to infinity here, you obviously get the left set number, because you obviously project on the harmonic forms, which represent the quorumology. And on the right-hand side, we make the limit as s then to 0. You obtain whatever formula there is for the left set number of g. So at the same time, this object remains constant and interpolates naturally between the left-hand side and the right-hand side. So this is exactly as the mental mechanism will apply in the case of the trace formula. So let me now take a compact manifold, x compact remanian manifold. Let me introduce delta x to be the Laplace-Baltrami operator on x. And then for fixed t, I just call g the heat current. So I deliberately use a segmentation that I used before for the nithiomorphism. So I'm going to ask four questions. So the first question is, is the trace of the heat kernel an Euler characteristic? What I mean by this is, can I think of the trace of the heat kernel as being an Euler characteristic of exactly the same type as before, just by doing the proper changes? Now, the second question is, can I express this Euler characteristic, or if you like, is generalizing the heat current using a much bigger object? So before, in the case of the chronology, we have the chronology, and we resolved it by the Durand complex. I'm asking, can I resolve in a certain way the space of smooth functions with values in R by a certain complex and re-express the trace here in terms of alternating some of traces on a much larger object? So the analogy again is passing from the chronology to the Durand complex. We want to pass here from the smooth functions to some complex which will resolve the smooth functions on X. And the third question will be, by making b tend to infinity, do we obtain Zellberg's trace formula? So in other words, can we think of the same type of trace formula as a generalized index formula? So again, I gave here the interpolation scheme. Here is a trace here on smooth functions of g of the heat kernel, which is a global invariant. By making b tend to infinity, we still don't know what the parameter is. We will obtain Zellberg's trace formula that we will think of as being a local formula. So what this means by analogy is that we will have to think of the smooth functions as the homology of some complex and even more as the harmonic objects in some complex. So I will try progressively to give the instruments which allow us to do this. So the answer is yes. So first of all, the question of resolving the state of smooth functions. So in the case of a manifold, again, we took the homology and we resolved it in the proper sense by the Durand complex. Here we're going to ask, can we think of the smooth functions on x as being the homology of something? So let me construct an infinite number of ways of doing this. So I just give myself a real vector bundle on x. And I call this script to be the total space of E. And then I introduce the space R. So this is the relative Durand complex. That the smooth forms are the total space, but which are just forms along the fiber. So this family of Durand complexes along the fiber. And DE is just a fiber-wise Durand complex. Now, I claim that my plurgery is lemma. The co-homology of this complex is precisely the smooth functions on the base. Well, this is because the co-homology of the Durand complex itself is made. Ultimately, it comes at a degree 0. It is a function which are constants, fiber-wise, meaning that they come from the base. So this is a complex whose co-homology is the smooth functions. Now, does this complex have a Hodge theory? Does this new complex R? Does it have a Hodge theory? So I claim that there is such a Hodge theory, because if you introduce GE to be an Euclidean metric, there is a Hodge theory for all these vector fibers. And this Hodge theory is in some sense given by the Wittner-Lapplation. So just a short reminder about Wittner-Lapplation. So in very simple case, I give myself E to be an Euclidean vector space. The classical obvious Hodge-Lapplation has continuous spectrum. But in some sense, the Wittner-Lapplation has discrete spectrum. So you can view it in one way, is to twist the Durand operator by the Gaussian function. Do not change the co-homology this way. So you construct the corresponding Hodge-Lapplation by taking the formula joint. What you get, essentially, is a Laplacian. You get a first operator, which is H, which is a harmonic oscillator, and N, which counts the degree in the exterior algebra. And so you recover, in some sense, the proper computation of the co-homology of a vector space, which is concentrated in degree 0, because H is the kernel of H is one dimensional, and N, 0 forms. So just to remark here that via the so-called Bergman's isomorphism, you can instead replace the Durand complex of E by the algebraic Durand complex on E. That is, this complex is quasi-isomorphic to this algebraic Durand complex. And then we see appearing here the polynomials, which are the symmetric algebra. And as we know, which is a natural inverse to the exterior algebra. So this construction here will play an important role in the cycle. So now we have our base manifold, X, and the fiber bundle, which is E. And the question will be to be how to couple the base and the fiber so that we can do something. So we need to introduce, first of all, the heat operator on the base, X, coupled with the written Laplacian on E. So this is the way that we'll approach the construction of this ultimately high-polyptical Laplacian, how to couple the base, which contains the heat kernel, with the fiber, which contains the written Laplacian. So let me explain the construction explicit in the case of a symmetric space. So let G be a real reductive group, K to be a maximal compact subgroup, and X equal G over K to be the symmetric space. I then introduce a corresponding carton splitting equipped with the non-generated bill in a form B. And this carton splitting descends to a vector bundle, E, which is made of two pieces, TX plus M. So it's a general fact that on a symmetric space or locally symmetric space, the tangent bundle has a natural auto complement, which is this thing which I call N. TX is modeled on B, and N is modeled on K. And actually, our interpolation will ultimately be built on the total space of such a vector space, of such a vector bundle. So we will explain how to couple X and TX plus M by lifting first everything to G multiplied by the realtor. So just an example in the case where G is a sum of R, K is S1, X is the hyper-alf-plane, and TX plus M is of dimension 3, so that the total dimension of fiber bundle is of dimension 5. So how are we going to construct this coupling? So Casimir and Kostalt. So first of all, on the reductive group, we have the Casimir operator, which is a second-order differential operator acting on the group, which is neither elliptic, neither hyperbolic. So it is, in some sense, positive in the p-direction and negative in the k-direction. That's a second-order differential operator, and which is in the center of the vector space. It is in the center of the enveloping algebra. Now I introduce the clefted algebra C hat of G, which is the clefted algebra of G minus B. And as we know, this clefted algebra acts on the exterior algebra of G stuff. So I also introduce U of G, which is the enveloping algebra. So if you like it, the algebra of differential operators of invariant differential operators of the group G, and the clefted Kostalt-deroc operator is a section of the clefted algebra tensed with the enveloping algebra. So it's a slight but significant modification of the construction of the classical deroc operator. It is constructed via the fundamental closed three-form of the group G, which I wrote here. And here is a short formula for it. You don't need to know this, so this is just, in some sense, the classical deroc operator, where the EIs are just differential operators acting on G. And here there is a cubic correction in terms of the clefted image of the three-form. Now the basic result of Kostalt is that the square of the Kostalt operator, with my conventions, is equal to minus the casimir. You could have another convention in which you would get the plus the casimir, plus some irrelevant constant. So what this means is that the casimir operator has a natural differential operator square root. And this we will exploit in the context of analysis. So let me observe that the Kostalt operator, what it acts upon, it acts on smooth functions on G with values in the exterior algebra of G star, while the casimir operator itself acts on the smooth functions on G with values in R. So in some sense we get a sort of straight root of the Laplacian, but it acts on the wrong space. So it's like straight root of minus one, which was just outside the real line. Kostalt, because I had K for, okay, I thought of this. I had K to be the maximal compact subgroup, and I did not want to have the question, what is K Kostalt? So this was made for you. Okay, so the operator Db now. So let me construct a sort of generalized Dirac operator. This Dirac operator will act on G product-ally algebra. So this new Dirac operator will act on smooth functions on G tensor, the types of the algebra, with values in lambda of G star. Okay, so, sorry, this is going too fast. So here is the formula. So I have a first term, which is just a Kostalt Dirac operator. So this one acts on the infinity of G tensor with lambda of G star. This is for the first piece. This one, you should not look at it. Absolutely not. Do as if it was not there. So this piece here is just a version of the Hodg-Durham written operator. That is, that's a sort of D plus D star, if you like, but in the proper Hodg theory. That is, in the proper Hodg theory of P and in the proper Hodg theory of K, which involves a written twist. Now you see that these two pieces, P and K, are treated differently. I mean, this different treatment reflects the fact that we are on a reductive group. But essentially, the effect when you square this is just to produce a very good Hodg-Lop-Lasch. So in some sense, these D papers, D per star, it acts on sin infinity of G of P, sorry, with coefficients in lambda of P star. And the other one acts on sin infinity of K with coefficients in lambda of K star. So if you combine all of these, you find that this new operator, D B, which I will call, for lack of a better name, a Dirac operator, acts on the smooth function of G cross G with values in lambda of G star. So what is the high-polyptical operation in this context? So what I'll do for the moment, I'm working on G, on the group G, 10 times the l-algebra G. So I take the square of the previous operator, D B square, and then I subtract from this square, the square of cos dot. So if you like, I do A plus B square minus A square. And here is a formula for this operator that I will explain, I will show to you. But before doing this, I descent, I quotient, the whole construction by K. So in other words, for the moment, we've done analysis on the group times the l-algebra, but then we have to descend everything to the symmetric space. So as I explained before, the l-algebra G descends to Tx plus m, G with its cut-out splitting, and G cross the l-algebra to the total space x hat of Tx plus m. So our new operator, lbx, which is obtained by descent, will act on the smooth functions on x hat, with values in the exterior algebra of T star x plus m star. So again, using the Fibromois-Bartmann isomorphism, instead of viewing the construction as acting on a bigger space, which is a total space of a vector bubble, you can instead view this operator as acting on an infinite-dimensional vector bubble. The infinite-dimensional vector bubble is just the symmetric algebra of T star x plus m star, multiplied by the exterior algebra of T star x plus m star. So here is the formula for this lbx. So this lbx contains along the fiber the harmonic oscillator of Tx plus m, but in some sense along the fiber it is just the written population. What we see appearing here as an operator which just differentiates in the base direction is actually the judizic flaw. And we have a bunch of other matrix terms that I will not explain. There is also this extra term here, which is a term of order four, which means that our operator, we are quitting the unshattered word of harmonic oscillators to go to something slightly more complicated of operators with coefficients quadratic, but this piece is absolutely necessary to make the theory work. So this operator is hyperliptic. So what does this mean hyperliptic here? That means that it does a good analysis in spite of the fact that it just differentiates twice in the fiber direction, but only differentiates once in the base direction. So analytically it is good, it is part of the class of operators concerned, considered by Hermandem. And so what I'm saying, and I will not show this, is that in the proper sense this operator deforms the original Casimir operator acting on X, or if you like deforms the original Laplacian. There is a collapsing phenomenon which is relatively subtle, which means that analytically splitting this operator deforms the original Laplacian. But now the wonderful thing is that when B tends to infinity, it will be the judizic flaw which will dominate. And so when B tends to infinity, this will force the localization on closed physics when you consider traces of it cones. So the case of locally symmetric spaces now, so let me take gamma to be a compact torsion free discrete subgroup of G, and then we consider the quotient of X by gamma, which is a compact locally symmetric space. So all the constructions I have done, so I have done constructions over G, then I descended these constructions over X, and now of course I can descend the construction to the quotient, because basically all the objects I introduced are properly equivariable. And so the fundamental identity is to say that the trace, in some sense of ordinary heat kernel, so Cz, in some sense you can think of this as a heat kernel, now Cz minus C, as being the, this is a sort of, I mean it's just modification of heat kernel by constant, is equal to the super trace of the corresponding hyperliptic operator. So exactly like an index theory where you say the original index, or Fshad number, is equal to the super trace of a heat operator acting on a bigger space, we have exactly the same sort of phenomenon here, and the proof is in this context rather elementary, it relies on the fact, on the collapsing of this operator to this one as B10 to zero, plus the invariance of the super trace under deformation by B, which is essentially an index theoretic proof. However this is not good enough for our purpose, because in some sense we're not going to apply to play the Zellberg's trace formula in its algebraic version. So actually when you have the trace of each kernel, locally symmetric space, you know that you can re-express it as a sum of objects which are evaluated on the symmetric space. So you can go back to the symmetric space. So in other words, in this equality, we can as well evaluate the left hand side and the right hand side as being sums, infinite sums, over the district root gamma. And now the true miracle is, is that it's not only true that the trace on the compact locally symmetric space is equal to the super trace again over the compact locally symmetric space, but orbital integral by orbital integral, there is a preservation. So now we're going to move to different sort of geometry. So the identity splits as an identity over orbital integrals, and this is why we will be able eventually to give geometric formula for that. So semi-simple orbital integrals. Let me take gamma to be semi-simple in G, and let me call gamma, right, it's country AC class in G. Then you can, this has already been introduced here, introduced in the proper sense, the orbital integral, in this case of the heat kernel, which is an integral of the quotient of G by the centralizer of gamma in the group. So in the proper sense, this integral makes sense, in spite of the fact that Pt of X is just a heat kernel of X over the symmetric space. So what I'm going to try to explain is how first of all to geometries the orbital integral, how to describe it as a geometric object. And as soon as we have done this, we will see that naturally the closed-index or minimized-index will come in the game even before doing any analysis. So given a semi-simple element gamma in the group G, I introduce the displacement function d of X and gamma X. So d being the remanent distance on the symmetric space. So it is a fundamental fact, a consequence of negative curvature or non-positive curvature, that this function d of d, the distance in general, is convex on X cross X. Convex in this case means that when restricted to due this X, it's convex. And so in particular, the function d of X to gamma X, a distance of X to gamma X, is convex. And the fact that gamma is semi-simple just says that the minimizing set for this displacement function is a convex subset, non-empty convex subset of X. So this is just the same simplicity of gamma which tells you this. So actually in our case, X of gamma, the minimizing set for the displacement function is actually the symmetric space for the centralizer z of gamma. So in other words, if you take a semi-simple element gamma and you look at the symmetric space associated with centralizer, the symmetric space itself is the total idio-disiq sum manifold of the manifold X. So how can we understand geometrically what is the orbital integral? So I wrote here X of gamma. So I take a judizik at X0 which is fixed, which is normal to X of gamma. And I take its image by gamma. So X of gamma is preserved by gamma, so this image is another judizik starting at gamma X0. Now because of negative curvature, these two judiziks are going to go far away. That is, the mutual distance of Y and gamma Y grows at least like the norm of Y, like the distance to X of gamma. So what is the orbital integral? The orbital integral is just the integral essentially of the heat color of X. Evaluate on Y and gamma Y, and you integrate just in all the possible normal directions with R of Y here being coefficient, which is a Jacobian. That's a geometric interpretation of the orbital integral. The fact that the orbital integral exists just uses the fact that the heat kernel at large distance decays like a Gaussian and compensates the possible growths of the decoding. So the proof we will give for the evaluation of this orbital integral will consist in pushing properly the integral to X of gamma by a deformation process. So let me now, well maybe I skip this, and let me give now a second fundamental identity, which is the fact that in effect the original orbital integral for the heat kernel, the elliptic one, the classical one, is given by the new orbital integral for the high-politic heat kernel. You can think of this as being an orbital integral for a nifidic dimensional vector bundle on the raise X. So these two orbital integrals are the same. And now we will make B tend to infinity in order to localize. It will localize to X gamma. So let me now explain the limit as B tend to infinity. So when you look at this high-politic Laplacian, you find that the dominating term as B tend to infinity is given by the jubizic flow after rescaling, so which forces in some sense the dynamics of the heat kernel to concentrate on closed-edizics. So let me now explain the final formula. So to explain the final formula, you can entirely forget about everything I said before. So let me take gamma to be a semi-sepal element, which I write in a canonical form after conjugation as a product of e to the a times k minus 1, where a is in B, k is in the group k, and a and k commute. Let me introduce z of gamma to be the centralizer of gamma, whose le algebra splits over the carton splitting as p of gamma plus k of gamma. So what is the formula for the semi-sepal orbital integrals, at least for the heat kernel? So the formula says the following. There is an explicit function, j gamma, which is evaluated on the k part of the le algebra, the centralizer of gamma. So you take this element gamma, you take this le algebra, you split it with p part and k part, and the orbital integral will be evaluated as an integral on the k part of the le algebra. So this is the orbital integral for the heat kernel. So this is an integral which comes from the geometry of x. And the right hand side is... Okay, so this exponential minus a square over 2t, this corresponds in some sense to the lengths of the due to the length squared associated with gamma, but above all what we have is an integral of k of gamma. And this integral of k of gamma, well there is a heat kernel here, it's not too bad. Here we have... Okay, I changed slightly gears because I introduced an homogeneous representation of k, and now I view my chasmir as acting on the sections of an extra vector volume. So this I did not tell you before, I was working before just with a lot of function functions. I also introduced now twisting, I just twist everything by representation of k. So we have a certain term which looks like a chasm character, like an equivalent chasm character. And so what when k easily believes that this is like a chasm character, this function j gamma will be like a thought form, or like an a-roof form. So note again the integral k gamma part, which is something which is in some sense invisible to classical analysis, it's somewhat ironic, that ultimately the final answer is expressed in terms of integral of k of gamma. So this formula accounts for the first term in Zellbrich's Trace formula I gave in the case of the main surfaces. So what is a function j gamma? It's a function again of the k part of the le algebra of the centralize of gamma. Now it looks very impressive, it is not. Well, I mean, I told you that we have this vector bound on tx plus m, sorry tx plus m. I would like to say that we have actually tx minus m. So this term j of gamma, it's a function of course, but it should be thought of as a ratio of the a-roof for tx divided by the a-roof of m. So it's really not like the classical Lefcet's formula, this tx plus m smithing propagates until the very end, and so we have the quotient here. So to make things more precise, in the case where gamma is trivial, if gamma is equal to 1, then the function j gamma of y0 is equal to the a-roof of i-ad of y0 acting on p divided by a-roof of i-ad of y0 acting on k. So in other words, instead like in classical Lefcet's formulas where we integrate differential forms, what we now integrate is functions under the algebra, and we integrate them whole, I mean, on the full k of gamma. So again the obvious analogy with a t-adbott, I mean, the formula, a fixed point formula of a t-adbott, is a pairing of a-roof of tx with a convention character of e. What we have here is just a sort of fake a t-adbott formula where of course the content is entirely different. And so tx in particular is replaced by tx minus m. Now, the last part is the connection with the loose space of x. So let x to be a remainder manifold and lx to be its loose space. So the loose space will be just loose maps from s1 to x at that stage. So as we know, lx is a remainder manifold, we can equip it naturally with the l2 metric, and s1 acts isometrically on lx, and the generator of the action is k of x, which is x dot. So in particular the zero set of this generator, and the trivial loops, they are point x, that's x, so x sits in lx as a zero set of a vector field, of a killing vector field, and a normal bundle to x in lx are just functions. You just take a point x, you take the tangent bundle, txx, and you look at all functions from s1 into this fiber whose integral is equal to zero. That's a normal bundle to x to lx. And now what I shall do is evaluate the inverse of the equivalent overclass of the normal bundle to lx in lx. Well, I mean, we are in the context where you could apply equivalent chronology, all these vector bundles are acted upon by s1. So the only difficulty is that since this normal bundle is infinite dimensional, we would have an infinite product. But if the infinite product is suitably normalized, this was a remark of Witton and Thierre that you get exactly the A-roof genius of tx. So in other words, the inverse of the equivalent overclass of the normal bundle is just given by a known characteristic class. So index theory and localization formulas. So I'm going back here to a Thierre Witton and later work I did on this, which is to say that you can interpret the heat equation formula. When you look at the heat equation formula for the index, it just looks like a magic analytic trick. Now, if you view this from a physicist's point of view, you can rewrite this formula at the first stage in a rigorous way as an integral of some measure. But actually, if you do the physicist's way, you just rewrite it as an integral of an explicit differential form. This differential form does not exist formally, but still it has a proper symmetry. It is essentially closed. It is equivariately closed under the action of s1. This was the observation of a Thierre that if you apply localization formulas in equivine commonality, what you get is a true answer, which is to say that the index of the x is equal to the integral of x of A-roof stancher. So it's as if the mechanism of equivine commonality was already at work in implicit form in index theory. So the form beta is here, the alpha t, maybe I should just explain what it is briefly. It is an explicit form which exists universally on any space with an action of s1. It contains as a leading term here essentially alpha t. It contains as a leading term exponential minus k squared over t, e to t. There is an extra term of degree 2. Bj is also explicit. It is a lift as an equivine form of the turn character form of e. And so if you apply formally without doing a limit, localization formulas in equivine commonality, you get the index formula. So at the time I was upset with this sort of argument, but later on I convinced myself that what the heat equation is doing to prove the index theorem is just doing a loop space, what you would do on any space. So in other words, it's not a loop that the analysis has invented some new procedure. It is the opposite which is true. It is just exploiting the best of its capabilities what exists universally on any space equipped with an action of a lead group. Now, let me go to orbital theory of the loop group. So let G be a compacted connected loop group, sorry, group, compacted connected lead group, not loop group. And let Lg, sorry, sorry, Lg, is its loop group that just moves maps from S1 into G. So I will not here introduce a central extension. I prefer to introduce the, for the moment, L till the G which is a semi-direct product of Lg with S1. We're taking into account that S1 acts naturally on Lg. So the le algebra of this extended lead group is now an algebra of differential operators on S1. So it is spanned linearly, the first piece which is d over dt, and the second piece which is at, where at is a periodic section of the le algebra. So actually, essentially the coidroid orbits of the extended loop group to respond to connections. So this I will explain with a homonymy lies in an orbit in the group. Well, because if you look at d over dt plus at, this is exactly a connection. So you can view in some sense the elements of the le algebra as being connections of the trivial G bundle. And the coidroid orbit corresponds to connections of the trivial G bundle, whose an anonymy lie in a conjugacy class in the orbit G. So if instead of looking at the connections, you look at its homonymy, you just integrate the, by the parallel transport for the connection. So in other words, you integrate the equation d over dt plus at of gt equal to 0, which is 0 equals 1. What you get is a pass which connects in the group, the identity to an orbit in the group. So in some sense, the coidroid orbits of the loop group are just the passes which connect the identity to an orbit in the group. So these are naturally symplatic metaphors. So heat current localization. So if I might compactly group by introducing a very metric, delta G is a fraction, and I can see that heat current on the loop group. So pt of g, just if you write it the way that physicists do, and I'm sure the way that you would do it, you get the pt of g is an integral over this orbit space of exponential minus d energy over t times dg. So I claim that this integral can be transformed into a Bellin-Vell integral. Because indeed, the energy of a pass corresponds just to the Hamiltonian for the natural action of s1 on the coidroid orbit. And so of course we don't know what dg is, but for lack of a better measure, we will pretend that dg is just a symplatic measure on the orbit. So if you apply again formally the localization formula as the equivalent of homology, just at the formal level, what you get is a formula which expresses pt of g as an integral over the fixed point sets of this action of s1 here, of the action of rotation, and that are just one parameter semi-groups. So actually you obtain again a known formula, you obtain a formula which is correct, which seems to say that the equivalent of homology again applies in this context. So let me give a remark by Igor Frankl and Atiyah. So Igor Frankl had observed that the heat kernel appears in the numerator of the left sets formula for the, well I should say the central extension of lg. So when you compute the character formulas, essentially what you have in the numerator is a heat kernel, and he suggested that this reflects a Kirinoff to left sets fixed point principle. Well pt of g again formally is an integral group space, and left sets formulas for character groups is just a sum over the affine line group. And if you look at this, in fact that these two things are equal, you see that at least formally this reflects a Kirinoff left set fixed point principle. So Atiyah suggested that the localization formula should apply as to the integral, and so what does the formal proof of localization will consist of? So how will we deal rigorously with this question of knowing does localization formula techniques apply to these integrals over the loop group? So if you look at what k is in this case, so the points of the point order of it are just connections, v over dt plus at, the vector field which consists in looking at the shift of the connection is just a dot, so that k squared is equal to the integral of a dot square, and so ultimately knowing that a is equal to g dot, what you see appearing is integral of g double dot square. So actually if you just formally try to report the message known to put localization formulas in equivocal ecology, you're unavoidably glad to look at such a pathogen rule which is a regularised pathogen rule, and which corresponds in the proper way to a hyperlinked So this I would expect. So the formal consequences, introducing the costant Dirac operator is a way to prove the Fourier formula for PTRG using localization formulas. In the context of symmetric space of non-compact time, the same mechanism can still be used but it's subtler. In other words, the whole question of descent from a reductive group to its compIc quotient to its quotient by k is not incorporated in the traditional mechanism of localization formulas. Okay, so here are just a few references, and that's it. Thank you very much. Thank you. Well, I mean, you know, if you have taken down the question of taking non-semi-semi-semi elements, of course, I mean, it's fundamental. But in some sense, that's a different question. That's a different question. That is the proper way to approach it is in some sense to approximate a non-semi-semi element by some simple element, truncate the interval, and then see what happens. This method will not, it will not delete the fundamental analytic and geometric difficulties they are when you go to the cusp, and in particular to iterated cusps where the structure of the boundary becomes very complicated. So this you will still have to work. You will discover them in a different language, and they will still be there. Thank you very much.
not a funny expansion. In this case, it's exactly plurgery is lemma. The co-homology of this appearing here the polynomials, which are the construction of this ultimately high-polyptical operator has a natural differential operator generalized Dirac operator. This Dirac operator here is a formula for this operator that I will phenomenon here, and the proof is in this context not only true that the trace on the compact eventually to give geometric formula for that. So semi-simple orbital integrals. Let me take gamma connections. So this I will explain with a sense, the coidroid orbits of the loop group are the context of symmetric space of non-compact in the traditional mechanism of localization