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?)

“Perverse” is very emphatically necessary here. G/B may be smooth, but the Schubert cells certainly aren’t.

Please expand on this interesting remark!

Cool… Did Kazhdan also say something about “cuspidal” unipotent representations? (I don’t actually know the definition offhand, I just remember that they are mysterious and sometimes exist — e.g. Sp_4 — and sometimes don’t — e.g GL_n.)

The existence of cuspidal unipotent representations is the whole point. Cuspidal representations are exactly those that don’t occur in parabolic inductions from smaller subgroups, and those that occur in functions on G/B are those parabolically induced from the trivial group. So Kazhdan was exactly pointing out an interesting way of getting at the cuspidals.

Maybe I should disown it instead of expanding on it; your comment

(Note to self: G/B is smooth and proper, so I’m not sure whether it’s necessary to say the word “perverse” here.) didn’t make any sense to me in the context of your post, and thus caused me to parse things a bit differently than I should have. Saying the “the derived category of l-adic sheaves” is perfectly fine, and that has absolutely nothing to do with G/B being smooth and proper. In fact, talking about “the derived category of perverse sheaves” after fixing a stratification will only lead you to grief (in my experience).

The point that was presumably what you were really remembering is that any sensible person will endow this category with its perverse t-structure, and that where all the action really happens in this story is in the category of perverse sheaves and its shifts in the derived category. The decomposition theorem tells you that under basically all sensible operations you want to do, direct sums of semi-simple perverse sheaves (shifted) are preserved, so everything can (and usually was) done just talking about sheaves of this form, rather than the whole derived category.

OK, I see… Thanks for clarifying!

Right, Kazhdan did say that the cuspidal representations were the “missing” ones — or rather, I thought he said this but it was rather confusing in my notes so I didn’t mention it in the post.

Thanks for the very nice post! Two recent papers directly related to this story are arXiv:0902.1493 by Bezrukavnikov, Finkelberg and Ostrik and arXiv:0904.1247 by Nadler and me (the latter is a representation theory cousin of the algebraic geometry paper with Francis you mentioned). The paper by BFO calculates the Drinfeld center of the abelian category of perverse sheaves on B\G/B and identifies it with Lusztig’s unipotent character sheaves, using this to complete a beautiful program by its authors to give a geometric approach to Lusztig’s classification of character sheaves.

I can (obviously) comment better on the second paper, which works in the derived context and over the complex numbers. The idea is to look at categorified representation theory of reductive groups, by which we mean module categories over appropriate categorified “group algebras” – in this case the “smooth group algebra” D(G) of D-modules on the group (which is a monoidal infinity-category). Rather than study this group algebra directly, we look at its subalgebras of B-biinvariant objects, the analog of the classical Hecke algebra you mention. In fact if we allow a little bit of twisting (given by fixing a local system on the torus, i.e. a parameter in the dual torus) we in some rough sense recover all the representation theory of D(G), but the most interesting part is the unipotent part D(B\G/B) (corresponding to D(G) modules generated by B-invariants).

One question we address is what are characters of such modules? There’s a very general answer to such questions — modules for any associative algebra object of any category [derived, or rather infinity, in our setting] the natural home for characters is Hochschild homology (which is the home of the universal trace on the algebra, a dual notion to Hochschild cohomology or center..in the context of monoidal categories, Hochschild cohomology is the (derived) Drinfeld center).

So a natural question is what is the Hochschild homology of the Hecke category?

We prove it coincides with Lusztig’s category of (unipotent) character sheaves, Lusztig’s geometric/categorified avatars for characters of finite groups of Lie type. In fact the Hecke category is in a precise sense the derived, categorified analog of a semisimple Frobenius algebra, which in particular implies that this coincides with the Drinfeld center/Hochschild cohomology (in the same way that class functions are both the center of a group algebra and its Hochschild homology, where characters live). In other words character sheaves ARE precisely the characters of categorified modules. (Moreover we find they carry all the structure of a topological field theory and agree for the group and its Langlands dual group, but that’s even more off topic than the rest of this egregious comment.)

It’s interesting to consider the most basic example of a character sheaf, the Springer sheaf (the character of the standard module D(G/B), analog of functions on G/B in your post). This sheaf (like any character sheaf) is a conjugation invariant system of differential equations on the group. In fact it’s precisely the system of equations that the distributional characters of infinite-dimensional representations (Harish Chandra modules) of the Lie group G satisfy, which Harish Chandra used to prove their remarkable regularity properties. This is the complex numbers analog of the function-sheaf dictionary you described — rather than pass to traces of Frobenius, we can look for solutions to our D-modules and again find ordinary characters of representations.

[Sorry this got out of hand in length..if only these comment boxes had word limits..]

David, Is it really “Drinfeld center of the abelian category of perverse sheaves on B\G/B” that’s being discussed in BFO? In the paper, it seems to say Harish-Chandra bimodules, which isn’t the same thing at all (for that you need N\G/N, with monodromic conditions on both sides).

Ben – of course you’re right, though not sure the difference appears on the resolution of this discussion.

A bit of context – Harish-Chandra bimodules are a mildly twisted form of perverse sheaves on B\G/B (or twisted D-modules on B\G/B with small twisting parameter). On the derived level the two versions (equivariant and monodromic variants of the Hecke category) are equivalent thanks to Beilinson-Ginzburg-Soergel, though that result perhaps ought to be viewed as relating Hecke categories for Langlands dual groups, and is the source of Langlands duality in our paper and an example of S-duality in the work of Ben and friends (Braden Licata and Proudfoot).

[...] Followup on my earlier claim to have been categorified. [...]