# Representations of reductive groups in characteristic p

I’ve been at a couple of interesting conferences lately and so I have a lot to talk about. I’ll start by summarizing an excellent expository talk by Jonathan Brundan which he gave at an MSRI introductory workshop last week.

Let G be a reductive group over an algebraically closed field k of characteristic p. The topic of this post is the algebraic representations of G. In other works, we want to study algebraic maps $G \rightarrow GL(V)$ where V is a finite dimensional vector space over k. Over the years, a few people (Soroosh, Carl, Alex Ghitza) have asked me what I knew about this theory and I’m afraid that I always gave them very incomplete or inaccurate answers. Now, that I’ve been to Brundan’s talk I think that I understand what is going on much better and I’d like to summarize it. Of course there will be nothing “new” in this post — I think that all the theory was worked out 20 years ago.

First, let us consider a construction of the group G. We start with our usual reductive Lie algebra $g_\mathbb{C}$ over the complex numbers. Consider its universal envelopping algebra $U_\mathbb{C}$. It has a Kostant-$\mathbb{Z}$ form, $U_\mathbb{Z}$, which is generated by $E_i^k/k!$ etc. Then tensor with k to get $U_k$ and dualize to get $k[G]$ the Hopf algebra of functions on G.

Now we can build Verma modules $M(\lambda)$ (ie modules over $U_k$) for $\lambda \in X(T)$ (the weight lattice), just as we do over $\mathbb{C}$. By general nonsense, $M(\lambda)$ has a unique irreducible quotient $L(\lambda)$.

Theorem
$L(\lambda)$ is finite-dimensional if and only if $\lambda$ is dominant.

The proof of this is quite interesting. The hard part is to show that $L(\lambda)$ is finite dimensional. It suffices to find at least one finite-dimensional $U_k$ module with highest weight $\lambda$. One way to do this is to start with the usual irrep $V(\lambda)_\mathbb{C}$ over the complex numbers. Then we have a $\mathbb{Z}$ form by acting on the highest weight vector with $U_\mathbb{Z}$. Then we tensor with k to get $V(\lambda)$ which is a finite dimensional highest weight $U_k$ module, called the Weyl module.

So for any dominant $\lambda$ we have two representations of G, $V(\lambda)$ and $L(\lambda)$ with highest weight $\lambda$. In fact $V(\lambda)$ is universal among such representations, since we have the following Borel-Weil theorem.

Theorem
$V(\lambda) = \Gamma(G/B, \mathcal{O}(-\lambda))^{\star}$

Here as over $\mathbb{C}$, the higher cohomology of these line bundles vanishes. Moreover, the character of $V(\lambda)$ is given by the Weyl character formula. The dimension of the weight spaces of $L(\lambda)$ will be smaller than that of $V(\lambda)$.

Sometimes $V(\lambda)$ and $L(\lambda)$ coincide. For example, $L((p^r -1)\rho) = V((p^r-1)\rho)$. These are called the rth Steinberg modules.

I don’t know if the characters of $L(\lambda)$ are known in general (is there an expert out there who can answer this question?), but there is the following remarkable theorem which reduces their study to that of a finite number of characters.

Since our group $G$ is defined over $\mathbb{F}_p$ (it is defined over $\mathbb{Z}$), we have a Frobenius map $F:G \rightarrow G$ which is a group homomorphism. We can use this Frobenius map to twist representations. It has the result of multiplying all the weights by p, and leaving representations irreducible, so we have $L(\lambda)^F \cong L(p\lambda)$. This has a remarkable generalization.

Theorem
Let $\lambda \in X(T)_+$ and suppose $\lambda = \lambda_0 + p \lambda_1 + \dots + p^k\lambda_k$. Then $L(\lambda) \cong L(\lambda_0) \otimes L(\lambda_1)^F \otimes \dots \otimes L(\lambda_k)^{F^k}$.

In particular, note that we can make such a decomposition of $\lambda$ with all $\lambda_i$ living in the region defined by the inequalities $0 \le \langle \lambda, \alpha^\vee \rangle \le p$. So to “know” all the irreducible representations, it is enough just to know those ones for $\lambda$ in this region.

What is remarkable about this theorem is that its proof involves considering the kernel of the Frobenius — a highly non-reductive group scheme which has just one k point.

Aside from thinking about these irreducibles, there is much more going on. The category is not semisimple, so it is not enough to understand just the irreducible objects. In fact there is a well-developed block theory which is related to the action of affine Weyl group on $X(T)$ generated by reflections in the fundamental alcove.

## 20 thoughts on “Representations of reductive groups in characteristic p”

1. Dear Joel,

last year I posted two articles on the arxive dealing with Lusztig’s conjecture for rational representations of reductive groups over a field $k$ of positive characteristic. The main result is the construction of a functor from a category of “special” sheaves of $k$-vector spaces on the complex affine flag manifold associated to the (simply connected) group $G$ to representations of the Lie algebra of the Langlands dual group $G^L$ over $k$.

The “special” sheaves are constructed from the skyscraper sheaf on the point Iwahori-orbit using integration along the fibres for the projection to partial affine flag
varieties. If $k=\mathbb C$, the decomposition theorem tells us that the special sheaves are direct sums of intersection cohomology complexes on affine Schubert
varieties. For the application to representation theory only finitely many special sheaves are needed and using a base change argument we can deduce that these are intersection cohomology complexes for almost all characteristics. This yields almost all instances of Lusztig’s conjecture.

The main idea of the construction of the functor is the following. The special sheaves on an affine flag manifold form a categorification of the affine Hecke algebra $H$, and the projective modules over the modular Lie algebra categorify its periodic module $M$. These categorifications are given by relating both sides to combinatorially defined categories. The first is a category of sheaves on an affine moment graph (which is by the way equivalent to the corresponding category of
Soergel’s bimodules), the second is a category appearing in the work of Andersen-Jantzen-Soergel.

Now letting $H$ act on the element $A_0\in M$ corresponding to the fundamental alcove yields a map $H\to M$. My functor simply categorifies this map using the
combinatorial categories above.

One of the advantages of this approach is that it directly relates sheaves of $k$-vector spaces to modular representations, without using the quantum group (i.e. the characteristic zero version). So it avoids the localization result of Kashiwara-Tanisaki and the Kazhdan-Lusztig equivalence. Moreover, it is, I believe, Koszul-dual to Lusztig’s program in the following sense. The above functor relates intersection cohomology complexes on the affine flag manifold associated to $G$ to projective representations of the Lie algebra of $G^L$, while Lusztig’s program associated (in case $k=\mathbb{C}$) to simple representations of the Lie algebra of $G$ to these sheaves.

However, the main problem remains: For a given group and a given field $k$ one still doesn’t know if Lusztig’s conjecture holds. In my paper “Multiplicity one results…” I determined the $p$-smooth locus of the affine moment graphs (by quite elementary, in particular non-topological arguments). This can be applied to Lusztig’s conjecture and yields its multipliciy one case in the following sense.

Lusztig’s conjecture can be translated into a conjecture about the Jordan-Hölder multiplicities of baby Verma modules: this number should be the same as the
corresponding periodic polynomial evaluated at 1. From the multiplicity one result for moment graphs it follows that if the latter number is 1, then also the multiplicity is one. This works for ALL prime numbers above the Coxeter number and is, as far as I know, the only result that holds in this generality.
Moreover, it is known that in general the Jordan-Hölder multiplicity is at least what Lusztig predicted. So the multiplicity one case provides, I believe, strong evidence towards the conjecture in general.

2. Joel,

The region you define is not the ”fundamental alcove” but the ”fundamental box” (think about A2). I believe the characters are known in the fundamental alcove (they are the same as in characteristic 0) but are not known in the fundamental box. As you point out, this would give character formulas for all simples. The Lusztig conjecture gives a conjecture for their characters (in terms of Kazhdan-Lusztig polynomials for the affine Weyl group) and this is known (by combining a few hundred pages of work due to Kashiwara-Tanisaki, Kazhdan-Lusztig and Andersen-Jantzen-Soergel) for “almost all p”. Recently Peter Fiebig has been able to give an alternative proof using moment graphs (which still uses Andersen-Jantzen-Soergel). Thus one knows that the Lusztig conjecture is “generically true”, however in any fixed characteristic one knows nothing.

3. Carl Mautner says:

Hi Joel!

First off, a good reference for all this is Jantzen’s `Representations of Algebraic Groups.’

The question about the characters is related to the non-semisimplicity of the category. The Lusztig conjecture predicts the characters of the simple representations L(V) for p greater or equal to the Coxeter number h, in terms of values of the Kazhdan-Lusztig polynomials. The conjecture has a number of strong implications, including a knowledge of all the Ext-groups between simples.

I’m not quite sure about the current status of the conjecture. I think Anderson-Jantzen-Soergel proved it for p sufficiently large, meaning there exists an N such that for all p greater than N the conjecture holds. Of course this doesn’t actually tell you whether or not the conjecture is true for any given p.

I have also heard that Bezrukavnikov has proven the conjecture for all p greater than the Coxeter number.

4. Indeed I went to a talk today by Lin on Lusztig’s conjecture. It is quite interesting, Lusztig’s conjecture tells you the character of L(\lambda) by telling you the multiplicities with which L(\lambda) occurs in V(\mu). Since you already know the characters of V(\mu), this determines the characters of L(\lambda).

At the end of his talk, Lin speculated on whether there could be a proof by geometric Satake. Indeed why not? You have an equivalence between G-reps and perverse sheaves on the affine Grassmannian and you want to prove that some multiplicities of G-reps matches some multiplicities of perverse sheaves on the affine flag variety (values of affine KL polynomials). So why not?

Final question: did Bezrukavnikov prove this Lusztig conjecture or a different one? With Arkhipov and Ginzburg, he proved the analogous Lusztig conjecture for quantum groups at a root of unity. (I think that you can go between the two Lusztig conjectures by the work of Anderson-Jantzen-Soergel).

5. Somehow it never occurred to me that you could get Ext information from the Grassmannian when your sheaf coefficients are in positive characteristic. It seems quite bizarre. What perverse sheaves correspond to $L(\lambda)$ and $V(\lambda)$?

6. Carl says:

Scott-

The IC extension from the orbit Gr^\lambda corresponds to the simple L(\lambda) and the perverse shriek extension to V(\lambda) (or maybe its dual? ugh, I always mix them up…).

7. shriek extension sounds right to me. That’s the one with a natural map to the IC sheaf.

8. On the other hand, the affine KL polynomial record Exts between perverse sheaves on the affine flag variety with coefficients in char 0 (I think that this is correct).

9. Outside of this geometric Satake situation, do you know if anyone studied perverse sheaves with coefficients in characteristic p?

It seems like it should have been studied by topologists since they are very interested in cohomology of spaces with coefficients in characteristic p and from there it is a short trip to constructible sheaves with coeff in char p.

10. Joel-

Soergel and his crew have studied them on the finite flag variety G/B, in which they also have an interpretation in terms of the representation theory of the algebraic group G over a field of characteristic p.

As I have been working on closely related questions, I have become very familiar with some of the difficulties. In particular, (a) one doesn’t have the usual notion of universal coefficients and (b) one doesn’t have the theory of weights, so spectral sequences don’t necessarily collapse and there is no decomposition theory. The reason people usually transfer problems into the language of perverse sheaves is to be able to use Deligne’s machinery of weights, e.g. in the proof of the Kazdan-Lusztig conjectures.

11. Carl –

what do you mean “one doesn’t have the usual notion of universal coefficients”? I have only recently started to think about these things!

Joel –

One should also mention Daniel Juteau’s thesis which defines a Springer correspondence “modulo l”. The idea is to relate modular representations of Weyl groups to perverse sheaves on the nilpotent cone with positive characteristic coefficients. An example of one of his results is that knowing the decomposition numbers for the symmetric group in characteristic p is equivalent to working out the characters of all the equivariant intersection cohomology complexes on the nilpotent cone with coefficients in characteristic p.

A beautiful example of this already occurs for S_2: in this case the nilpotent cone is a quadric cone in affine 3 space and the intersection cohomology complex looks different in characteristic 2 to all other characteristics (all other characteristics thinks the cone is smooth!)
Under Daniel’s translation, this becomes a very complicated way of saying that the representation theory of S_2 is different in characteristic 2 to any other characteristic !!

12. Geordie says:

Carl —

I just spoke to Peter and now I understand what you were saying with point a).

The point is that if you know the intersection cohomology of a variety with coefficients in Z, then you do not necessarily know it over a finite field for example. The reduction of the intersection cohomology complex mod some prime isn’t necessarily the intersection cohomology complex. (This is also what Daniel talks about alot in his thesis).

13. Peter – Thanks for letting us know about your interesting preprint. I took a look at them. I think that it is great how you are able to use the moment graph theory. Do you know if there is any connection between your work and the geometric Satake correspondence which relates rational representations of G over k to sheaves of k vector spaces on the affine Grassmannian?

Geordie – For some reason, until now I had not seen your first comment. Thanks for drawing our attention to Peter’s paper and also for correcting my mistake about the fundamental alcove vs. fundamental box (I also noticed the mistake and corrected it the day after the post).

14. Joel – Geordie’s comment was in the spam filter. If he hadn’t complained to me about it, it might never have been found.

15. Geordie-

Sorry, I didn’t notice that the discussion had continued…

That is what I meant. I just came across Daniel Juteau’s thesis and started looking over it. Looks really interesting.

I’ve been thinking about trying to do something with moment graphs along the lines of a paper of Braden-MacPherson only with coefficients in positive characteristic. In Braden-MacPherson, they give a method for computing the stalks of the IC sheaf with char 0 coefficients from the moment graph. I was hoping that one could modify their procedure to compute the char p stalks in the affine Grassmannian or at least some vanishing of the stalks to give a proof of the linkage principle via geometric satake. So far, I haven’t made much progress and have found myself drifting towards just thinking about the integral IC stalks.

16. David Ben-Zvi says:

Hi,

I was under the impression (as Carl says) that Bezrukavnikov has a proof of the Lusztig conjecture for p>Coxeter number. This is not quite stated in his ICM but I believe he has said so at talks as far
back as 2002 and a strategy towards this is I think implicit in the ICM address (of course Peter’s result has the serious

– describing modular representations of Lie algebras in terms
of the derived category of coherent sheaves on the T^*G/B, the Springer resolution (joint with
Mirkovic and Rumynin)

– a derived equivalence between coherent sheaves on
T^* G/B and a quotient of the category of
perverse sheaves on affine flags (with Arkhipov)

– an enhanced version of the above, to an equivalence of
the SYMMETRIES of the above categories — ie coherent
sheaves on Steinberg and perverse sheaves on flags — which
are both categorified forms of the affine Hecke algebra

– a precise understanding of how t-structures behave under
all the above derived equivalences (with Mirkovic)

– finally (and probably the hardest and least written-up part?)
an understanding of Hodge structures (or mixed structures)
on all the relevant categories.

I don’t know how exactly this is supposed to add up,
but various other conjectures of Lusztig (including
the one Joel mentions, with Arkhipov-Ginzburg,
and various ones on character sheaves and two-sided cells
and related structures, with Ostrik and Finkelberg)
have fallen on Roma’s way towards this.
(As you can tell I’m a big fan! )

One thing I particularly like is that the Bezrukavnikov-Mirkovic-Rumynin picture gives a clear
conceptual picture how&why the affine Hecke algebra
and Springer theory appear in the modular representation
theory (though the appearance of the affine Weyl group is
clear, I think this was somewhat mysterious, at least
according to Humphreys’ Bull AMS article — but maybe Peter
can correct me here, as elsewhere!)

Peter: It would be extremely interesting to hear what points
of contact you can see between your beautiful recent
preprints and the geometry that I mention here, parallels
and differences etc — it’s exciting that between the approaches
a deeper understanding seems to be emerging!

(It’s also worth mentioning that Kremnizer has
shown that the same geometry — in particular
the “exotic t-structure” on T^* flags — controls
also the representation theory of quantum groups at roots
of unity, giving a clearer picture of the
results of Arkhipov-Bezrukavnikov-Ginzburg that Joel mentions,
while Gaitsgory-Lurie are giving a clean understanding
of the relation between quantum groups and loop groups,
so that all legs of the amazing Lusztig triangle
modular-quantum-loop are close to becoming
conceptually understood!)

17. David – Thanks for a nice summary of Roman’s work. I am a bit confused though. I thought that the BMR work was about an analogue of BB localization in characteristic p. In particular, it deals with category O for the Lie algebra g.

How does it relate to the rational representations of the group? Does that category sit inside the category O?

I thought that in characteristic p, there wasn’t such a strong connection between representations of the group and those of the Lie algebra — in particular whenever the group acts, then the divided powers (like E^k/k!) should also act which are not in the Lie algebra.

18. David Ben-Zvi says:

Joel,

The group representations correspond (as far as I know —
please correct me all the experts out there!) to “restricted”
representations of the Lie algebra. In the enveloping algebra
Ug we have a new center in characteristic p, the p-center,
and we have to set it to act by zero to get representations
that integrate to the group. The BMR picture
treats all reps of the Lie algebra, in particular the
restricted ones. (I think the analogue
on the other side of the Langlands type correspondence
is that the group corresponds to affine Grassmannian
mod spherical subgroup while the Lie algebra
corresponds to Grassmannian mod Iwahori — or more
precisely to a category of the same size, the
Iwahori-Whittaker category).

19. sorry for replying so late – I travelled to Aarhus last weekend.

Joel – the geometric Satake equivalence is quite different from the relation explained in my paper. First of all, geometric Satake gives representations of the group, whereas I obtain representations of its Lie algebra. Moreover, geometric Satake gives all representations, whereas I only get projective representations in the principal block. Finally, geometric Satake uses perverse sheaves constructible along the G[[t]]-orbits, whereas the “special” sheaves that appear in my paper are constructible along the smaller Iwahori-orbits.

Recently I discussed a possible relation between these two pictures with Geordie. The main idea (rather: the main speculation) is to consider certain representations of the group as acting on special sheaves by convolution, and on the modules of its Lie algebra by some fancy tensor product. The functor that appears in my paper should then intertwine these two actions.

David – thank you for the nice overview on Bezrukavnikov’s picture. I have to admit that I cannot see yet the precise connection to the results in my paper. But I believe the following: There are always two possible “localizations” of representation theories on flag varieties. The first is the Beilinson-Bernstein picture, the other is the Andersen-Jantzen-Soergel picture. They are mutually Koszul-dual in the following sense: While the first typically relates simple perverse sheaves to simple representations, the latter
relates simple perverse sheaves to projective representations. Moreover, the flag variety in the AJS-picture is also not associated to the Lie algebra in question, but to its Langlands dual. Taken together, both localizations yield a Koszul-self duality of the representation theoretic categories.

If I am not mistaken, Bezrukavnikov’s theory is about a Beilinson-Bernstein localization in the modular case. My paper is on the side of the AJS-philosophy. This is why I hope that one can get a Koszul-duality for quantum groups (and for modular representations) from comparing the two pictures.