I had the good luck to be in New York on Friday when David Kazhdan gave an unscheduled lecture at NYU about categorification and representations of finite groups. For people like me, who spend most of our days dismally uncategorified, the talk was a beautiful advertisement for categorification.
Actually, the first twenty minutes of the talk were apparently a beautiful advertisement for the Langlands program, but I got lost coming from the train and missed these. As a result, I don’t know whether the results described below are due to Kazhdan, Kazhdan + collaborators, or someone else entirely. And I missed some definitions — but I think I can transmit Kazhdan’s point even without knowing them. You be the judge.
It went something like this:
Let G be a reductive split group over a finite field k and B a Borel. Then C[G(k)/B(k)] is a representation of G(k), each of whose irreducible constituents is a unipotent representation of G(k). (Note: the definition of “unipotent representation” is one that I missed but it comes from Deligne-Lusztig theory.)
When G = GL_n, all unipotent representations of G appear in C[G(k)/B(k)], so this procedure gives a very clean classification of unipotent representations — they are precisely the constituents of C[G(k)/B(k)]. Equivalently, they are the direct summands of the center of the Hecke algebra C[B(k) \G(k) / B(k)]. For more general G (e.g. Sp_6, E_8) this isn’t the case. Some unipotent representations are missing from C[G(k)/B(k)]!
Where are they?
One category-level up, naturally.
(see what I did there?)
OK, so: instead of C[B(k)\G(k)/B(k)], which is the algebra of B(k)-invariant functions on G(k)/B(k), let’s consider H, the category of B-invariant perverse l-adic sheaves on G/B. (Update: Ben Webster explained that I didn’t need to say “derived” here — Kazhdan was literally talking about the abelian category of perverse sheaves.) This is supposed to be an algebra (H is for “Hecke”) and indeed we have a convolution, which makes H into a monoidal category.
Now all we have to do is compute the center of the category H. And what we should mean by this is the Drinfeld center Z(H). Just as the center of an algebra has more structure than the algebra structure — it is a commutative algebra! — the Drinfeld center of a monoidal category has more structure than a monoidal category — it is a braided monoidal category. It’s Grothendieck Group K_0(Z(H)) (if you like, its decategorification) is just a plain old commutative algebra.
Now you might think that if you categorify C[B(k)\G(k)/B(k)], and then take the (Drinfeld) center, and then decategorify, you would get back the center of C[B(k)\G(k)/B(k)].
But you don’t! You get something bigger — and the bigger algebra breaks up into direct summands which are naturally identified with the whole set of unipotent representations of G(k).
How can we get irreducible characters of G(k) out of Z(H)? This is the function-sheaf correspondence – for each object F of Z(H), and each point x of G(k), you get a number by evaluating the trace of Frobenius on the stalk of F at x. This evidently yields a map from the Grothendieck group K_0(Z(H)) to characters of G(k).
To sum up: the natural representation C[G(k)/B(k)] sometimes sees the whole unipotent representation theory of G(k), but sometimes doesn’t. When it doesn’t, it’s somewhat confusing to understand which representations it misses, and why. But in Kazhdan’s view this is an artifact of working in the Grothendieck group of the thing instead of the thing itself, the monoidal category H, which, from its higher categorical perch, sees everything.
(I feel like the recent paper of Ben-Zvi, Francis and Nadler must have something to do with this post — experts?)