Can we count the number of simple modules for a reduced enveloping algebra?

Question

Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple $\mathfrak{g}$-modules belongs to precisely one of the reduced enveloping algebras $U_\chi(\mathfrak{g})$ with $\chi \in \mathfrak{g}^*$. Thanks to a theorem of Kac–Weisfeiler the simple $\mathfrak{g}$-modules may be entirely understood by focusing on the case where $\chi$ is nilpotent, i.e. $\chi$ vanishes on some Borel subalgebra. Assume $\chi$ is nilpotent, let $\mathcal{B}$ denote the flag variety of $G$, and $\mathcal{B}_\chi$ the Springer fibre over $\chi$. As a set this is just the collection of all Borel subalgebras of $\mathfrak{g}$ where $\chi$ vanishes.

Now let $p > 2h - 2$ where $h$ is the Coxeter number. In a famous paper by Bezrukavnikov, Mirković and Rumynin the authors proved (amongst other things) a certain conjecture of Lusztig, which asserts that the number of simple $U_\chi(\mathfrak{g})$-modules with trivial central character is precisely the rank of the Grothendieck group of the coherent sheaves on $\mathcal{B}_\chi$. By a later theorem in that same paper this number is actually equal to the sum of the dimensions of the $\ell$-adic cohomology of $\mathcal{B}_\chi$. It is well known that the blocks of $U_\chi(\mathfrak{g})$ are determined by their central character, and after Jantzen's results on translation functors (see Proposition B5 in Jantzen's paper listed below) it is also known that each block contains the same number of simple modules.

My question is the following: is there a known formula for the dimensions of the cohomology of the Springer fibre over a nilpotent element? Has somebody used this to write down a formula for the number of simple $U_\chi(\mathfrak{g})$-modules? Using the results listed above is there anything interesting/unexpected/useful which we can learn about the category of $U_\chi(\mathfrak{g})$-modules?

When $\chi$ has standard Levi type the simple $U_\chi(\mathfrak{g})$-modules are classified by a theorem of Friedlander and Parshall. The classification shows that simple modules correspond to the orbits of a certain subgroup of the Weyl group (depending on $\chi$) on an $\mathbb{F}_p$-lattice of the chosen torus, and this leads to a formula for the number of simple modules. It would be wonderful if there existed a formula outside standard Levi type which generalises this, but it would also be interesting to see a list of numbers corresponding to Bala–Carter labels for nilpotent orbits in exceptional Lie algebras, for instance.

References:

R. Bezrukavnikov, I. Mirkovic & D. Rumynin, “Localization of modules for a semisimple Lie algebra in prime characteristic”, Annals Math., Vol. 167 (2008), 945–991.

E. Friedlander & B. Parshall, “Modular representation theory of Lie algebras”, Amer. J. Math., Vol. 110 (1988), 1055–1093.

J. C. Jantzen, “Subregular nilpotent representations of Lie algebras in prime characteristic”, Rep. Theory, Vol. 3, (1999), 153–222.

Answer 1

The answers to your several closely related questions are not yet known, though many parts of the story have emerged. In particular, there is no "formula" for the number of simple $U_\chi(\mathfrak{g})$-modules in a typical (meaning regular) block, and such a formula probably doesn't exist. It's quite difficult in general to compute the total dimension of the cohomology of a Springer fiber (which tends to coincide with the Euler characteristic, due to vanishing of the odd-dimensional cohomology). But as remarked in the question, in the special case when $\chi$ is of standard Levi type, the older work of Friedlander-Parshall does give a straightfroward answer: the quotient of the Weyl group order by the order of a parabolic subgroup.

The small example of type $G_2$ is instructive: in the paper by Jantzen which you cite, all of his ingenious algebraic methods couldn't quite determine the simple modules here when $\chi$ is "subregular" nilpotent. Uncertainty arose because he could construct two modules along with three others having equal dimension which might or might not be non-isomorphic. But the more geometric approach in the Annals paper by Bezrukavnikov–Mirković–Rumynin (where some of Lusztig's conjectures are proved) arrives at the number 5: here the Springer fiber is a Dynkin curve involving three parallel projective lines along with a fourth such line intersecting all three, and its classical or $\ell$-adic cohomology therefore has total dimension $5 = 4+1$. (The arXiv version of the 2008 BMR paper is here.)

It's worth adding that a 1983 conjecture by Lusztig still remains open: see 3.6 of Lusztig - Some examples of square integrable representations of semisimple $p$-adic groups. This proposes to count the number of one-sided cells in a given two-sided cell of an affine Weyl group (corresponding to a nilpotent orbit under his subtle bijection), by computing the dimension of the fixed points of a finite component group on the cohomology of a suitable Springer fiber. This in turn is closely related (by BMR) to the number of simple modules in a regular block considered here; in fact it coincides with that number when the nilpotent $\chi$ has a connected centralizer in the algebraic group.

For a survey with many references (as of about 1997), see my 1998 survey “Modular representations of simple Lie algebras” in the AMS Bulletin linked on my homepage here. Note that the conjecture in $\S19$ was later proved by Brown–Gordon in The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras. But the more speculative comments in the last paragraph of that section need to be revisited, in view of the results of BMR. It remains tempting to conjecture that the number of non-isomorphic simple modules in any block of a reduced enveloping algebra is bounded above by the order of the Weyl group. But this number certainly isn't always a divisor of that order, as $G_2$ shows.

For summaries of what Jantzen and I later worked out about the Lie algebras of simple algebraic groups of types $C_3$ and $D_4$ see the first two unpublished notes in that same list. Examples like these make it clear that explicit formulas for dimensions of simples, as well as for the total number of them in a regular block, may be out of reach. But the examples also reveal intriguing questions about $p$-divisibility of dimensions, going beyond the basic ideas of Kac–Weisfeiler and Premet.
[Note also the third unpublished note on my list, where I point out how a proof of Lusztig's 1983 conjecture would resolve a newer conjecture by Shi on counting one-sided cells.]

This is getting long, but I should mention a couple of imprecisions in the question. In order to identify nilpotent orbits in $\mathfrak{g}$ with those in $\mathfrak{g}^*$, one has to identify these spaces via a suitable nondegenerate form such as the Killing form (when such a form exists). So it's best to focus here on semisimple groups and their Lie algebras, or just on simple algebraic groups. Also, not all blocks of $U_\chi(\mathfrak{g})$ with $\chi$ nilpotent have the same number of simple modules, but the regular ones do, and these predominate as $p$ grows. (A smaller correction is the English spelling of the name Weisfeiler.)

ADDED 2019: It's worth noting that, as a follow-up to work of [BMR], a recent student of Lusztig (now a postdoc at Minnesota) named Dongkwan Kim, has worked out the Euler characteristic (usually the total dimension) of a Springer fiber for each type of root shstem. See for example his paper in various pereprint forms Kim - On total Springer representations for classical types and others listed there. But this is an algorithm rather than a "formula".

Answer 2

In the case when $\mathfrak{g}$ is a finite-dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$, the number of isomorphism classes of simple $U_{\chi}(L)$-module is bounded above by $p^{MT(L)}$, where $MT(L)$ denotes the maximal dimension of a torus in $L$. See Section 3 in the following joint paper with J. Feldvoss and Th. Weigel: http://link.springer.com/article/10.1007%2Fs00031-015-9362-5