I’ve recently been reading a paper which ties together a number of this blog’s themes: Canonical Quantization of Symplectic Vector Spaces over Finite Fields by Gurevich and Hadani. I’m going to try to write an introduction to this paper, in order to motivate you all to look at it. It really has something for everyone: symplectic vector spaces, analogies to physics, Fourier transforms, representation theory of finite groups, gauss sums, perverse sheaves and, yes, functions. In a later paper, together with Roger Howe, the authors use these methods to prove the law of quadratic reciprocity and to compute the sign of the Gauss sum. For the experts, Gurevich and Hadani’s result can be summarized as follows: they provide a conceptual explanation of why there is no analgoue of the metaplectic group over a finite field. Not an expert? Keep reading!
Let’s start with some simple “physics”. Let be the vector space of complex-valued functions on the real line. (We probably want to impose some conditions on these functions, but this is (1) a blog post (2) “physics” and (3) only for motivation. I can think of no better time to ignore analytic issues.) There are two important operators on : the operator , which sends to and the operator which sends to . They obey the relation . Basically all of the math in an introductory quantum mechanics course is playing with these operators.
For our purposes, we would rather exponentiate these operators. The exponential is the operator which takes to and the exponential is the operator which takes to . (If you haven’t seen this before, it is worth thinking about.) These exponentiated operators obey the relation
It is nice to build an abstract group to capture the formal properties of the above relation. The Heisenberg group is the group of formal symbols
where is a complex number of norm and and are real numbers. Multiplication is determined by the following relations:
For any in , the element is central.
So is a representation of the Heisenberg group. The Stone-von Neumann Theorem roughly states that is, up to isomorphism, the only irreducible representation of the Heisenberg group where the symbol acts by multiplication by the scalar .
Let be the vector space of formal linear combinations , with the skew symmetric bilinear form . Then we can more concisely define the Heisenberg group by saying that it is the group of symbols , with and , given the relations that is central and
Mathematical physicists like to work with , without choosing a splitting of into transverse subspaces and . Note that the definition of the Heisenberg group respects that aesthetic, but the definition of does not. For future reference, we set and .
An interruption: all of this works equally well for functions on a vector space of dimension larger than . I find that right about now is the transition point where that added generality becomes illuminating, rather than confusing. You might want to go back and rework everything for functions on a dimensional space. Note that, if you do things correctly, and will naturally be dual vector spaces. I’ll keep writing in the one dimensional case, then switch to the general case when I talk about what is new in Gurevich and Hadani.
Here is a definition of the Heisenberg group which uses the subspace but not the subspace . Let denote the abelian subgroup of the Heisenberg group consisting of expressions where . The quotient of the Heisenberg group by is canonically isomorphic to . Rather than realize as functions on , we will realize it as functions on the quotient of the Heisenberg group by . The language of induced representations lets us do this in a very concise way: is (canonically isomorphic to) the induction from of the obvious one dimensional representation of . By “the obvious one dimensional representation”, I mean .
For any one dimensional subspace of , we can define the group . Let be the induction to the Heisenberg group of the character of . (If you are doing the -dimensional generalization, then should be dimensional and should be zero, i.e., should be Lagrangian.) All of the representations are isomorphic, but not canonically so.
To get some intuition, let’s look at and . To find an isomorphism, we must find a way to translate functions on to functions on the dual space . The Fourier transform,
is such an isomorphism, and the standard fact that Fourier transform switches and turns into the fact that this is a map of representations. That is the constant I always forget, you know, the one that is something like . The issue of this constant is actually very important for us.
For almost any two subspaces and , we can similarly define a map to by
where we use the canonical identifications between and functions on . (Why “almost any”? Stay tuned!) We hit a technical point: what do I mean by ? I need to choose a volume form on in order to take the integral. So, properly, we should work not with subspaces and but with subspaces equipped with a nondegenerate volume form.
But now something strange happens! Even once the issue about volume forms is straightened out, there is no compatible continuous way to choose the constants . The space of one dimensional spaces of , equipped with volume forms, is a Mobius strip. Requiring that form to be nondegenerate cuts the strip down the center, leaving a cylinder. If you travel around that cylinder and come back to where you started, you’ll find you’ve picked up a sign error!
I’m sure there is a lot to say here about connections, topology and so forth. But I want to explain what happens in Gurevich and Hadani’s work.
Instead of working with , they work over the finite field for an odd prime. Fix a nontrivial root of unity . Let be a -vector space of dimension , equipped with a perfect skew symmetric form . Define the discrete Heinsenberg group to be the group of formal symbols , with , subject to the relations that is central and
For any Lagrangian subspace of , we can define as before. If is any lagrangian subspace transverse to , we can identify with complex-valued functions on . If and are transverse, there is an isomorphism by
(If and are not transverse, we can’t identify with . This is why I said “almost any” two subspaces above. We’ll talk more about this issue later.)
Now, once again, we need to think carefully about volume forms. (And this time, it is less obvious why, because there is a standard counting measure on .) However, once that issue is straightened out, a miracle happens: this time, we can choose the constants so that everything is compatible. In particular, the group of automorphisms of preserving acts on , even though the analgous statement for is false.
My understanding is that the above was known before Gurevich and Hadani, but their explanation, and in particular an explicit recipe for choosing the constants , was not. One of the innovations of their paper is that, using perverse sheaves, they extend the Fourier transform to the case where and are not transverse. Also, while I do not follow their proof yet, it seems to involve a number of identities between Gauss sums, so the number theorists should like this too.
A few questions that came to mind reading their paper: (1) Does this result say that the space of lagrangian subspaces of is simply connected? (A statement which is false for .) If not, is there some similar statement about characteristic topology which is true and give a conceptual explanation for this?
(2) Gurevich and Hadani work very hard to handle the nontransverse case. Is this necessary? Explicitly, consider this simplicial complex whose vertices are lagrangian subspaces equipped with a non-degnerate volume form and whose faces are pairwise transverse collections of lagrangians. Is this complex connected and simply connected? If so, then we could give an alternate definition of by composing the definition for the transverse case. Once we checked that going around a triangle in both ways gave the same result, we would know that our map was well defined. This certainly works when !