This is something which I suspect is written up in introductory books on mathematical physics if I knew where to look. Suppose I have some parameters $t\_1$, ..., $t\_k$ ranging over a neighborhood in $\mathbb{R}^k$. I also have $k$ matrix-valued functions of the $t$'s: $H\_1(t\_1, \ldots, t\_k)$, ... $H\_k(t\_1, \ldots, t\_k)$. These obey both
$$[H\_i, H\_j]=0 \quad (\ast)$$
and
$$[\partial\_i+H\_i, \partial\_j+H\_j] =0 \quad (\dagger).$$
For those who don't like the language of connections, we can expand $(\dagger)$ as $\partial H\_i/\partial t\_j - \partial H\_j/\partial t\_i + [H\_i, H\_j]=0$ or, in the presence of $(\ast)$, as 
$$\frac{\partial H\_i}{\partial t\_j} = \frac{\partial H\_j}{\partial t\_i}.$$


Equation $(\ast)$ tells us that, assuming the $H\_i$ are individually diagonalizable, we can find $u(t)$ a simultaneous eigenvector for all the $H\_i$:
$$H\_i(t) u(t) = \lambda\_i(t) u(t). \quad (\ast\ast)$$


Equation $(\dagger)$ tells us that the vector-valued PDE 
$$\frac{\partial v}{\partial t\_i} + H\_i v=0 \quad (\dagger \dagger)$$
will have a unique solution $v(t\_1, t\_2, \ldots, t\_n)$ for any initial value.


I'm pretty sure there is supposed to be a relation between the solutions to $(\ast \ast)$ and $(\dagger \dagger)$. What is the right statement, and what is the keyword to read about this situation?


Motivation: I'm trying to work through the papers of Varchenko, Scherbak and others on the KZ equation. I think it would really clear my head to just see this scenario described abstractly without all the details of which operators they are thinking about.


$\def\mg{\mathfrak{gl}\_n}$ Edit to spell out the relation. Let $V\_1$, $V\_2$, ..., $V\_n$ be representations of $\mg$. So $U(\mg)^{\otimes n}$ acts on $V\_1 \otimes V\_2 \otimes \cdots \otimes V\_n$. Let $\Omega \in U(\mg) \otimes U(\mg)$ be the Casimir. (Note: The element I learned to call the Casimir was a central element $c$ in $U(g)$. In terms of that element, $\Omega = \Delta(c) - c \otimes 1 - 1 \otimes c$.) Let $\Omega\_{ij}$ be $\Omega$ acting in positions $i$ and $j$.


For generic parameters $z\_1$, ..., $z\_n$, define $H\_i = \sum\_{j \neq i} \Omega\_{ij}/(z\_i-z\_j)$. Then, as I understand it, the KZ equation is $(\partial\_i + H\_i) v(z\_1, \ldots, z\_n)=0$, where $v$ is a function valued in $V\_1 \otimes V\_2 \otimes \cdots \otimes V\_n$. The $H\_i$'s obey both $(\ast)$ and $(\dagger)$ (a nice exercise). And people seem to be very interested in solving both $(\ast \ast)$ "diagonalizing the action of the Gaudin subalgebra" and $(\dagger \dagger)$ "solving the KZ equation". So I was hoping to understand how they relate, and why.



Hamiltonians which commute both as operators and as connections
Hi David,


I think there is indeed a relation, which I learned precisely from papers of Varchenko among others. All of this is rather classical and can be found e.g. in Etingof-Frenkel-Kirilov book "Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations". 


The fact that the $H\_i$ satisfies this stronger condition is equivalent to say that for any parameter $\kappa$ the operators $\kappa \partial\_i+H\_i$ alos satisfies ($\dagger$).


Hence you can take asymptotic expansion of solutions at $\kappa \rightarrow 0$ on some neighbourhood $D$ of some $z\_0$, of the form
$$e^{S(z)/\kappa} (f\_0(z)+O(\kappa))$$


where $S$ is a scalar valued function. Then you can show that, assuming the $H\_i(z)$ are simultaneously digonalizable then $f\_0$ is a common eigenvector of them, with eigenvalues $\partial\_i S$. Conversly given a common eigenvector at some $z\_0$ you can construct an asymptotic solution. So the usual trick, widely used in the study of the KZ equation, is to also take some asymptotic limit w.r.t. the variable $z\_i$ in such a way that eigenvectors are "easy" to find. The standard example in the KZ case is the asymptotic zone 


$$|z\_i-z\_1| \ll |z\_j -z\_1|\quad if\quad i < j $$


for which, up to some change of variable, the equation can be written
$$\kappa \partial\_i f= \left ( \Omega\_i/u\_i +reg\right)f\quad i=1\dots n-1$$
where $\Omega\_i=\sum\_{k < i} \Omega\_{k,i+1}$ and $reg$ is regular at $u=0$. 
Then given some common eigenvector $v$ of the $\Omega\_i$ with eigenvalues $\mu\_i$ there exists a unique solution of the form
$$(\prod u\_i^{\mu\_i/\kappa})(v+r(u))$$
where $r(u)$ is regular at $u=0$ and $r(0)=0$.


I'm not very familiar with D-modules (and by the way I would be happy is someone extends on this), but you can rephrase it as follows: viewing $\kappa$ as a formal variable leads to a filtration on the algebra of differential operators on $V$ (the vector space acted on by the $H\_i$) which in turn is nothing but the usual filtration by the degree of differential operators. Taking the associated graded turns the equation


$$(\partial\_i+H\_i)f=0$$


into the equation


$$(y\_i+H\_i)f=0$$


Whose solutions are clearly commons eigenvectors of $H\_i$. So I'm rather confident that you can say that the spectrum of the $H\_i$ for all common eigenvectors is the characteristic variety of the D-module of solutions of the differential equation you started with.


