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--Weisfeiller the simple $\mathfrak{g}$-modules may be entirely understood by focusing on the case where $\chi$ is nilpotent, ie. $\chi$ 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 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

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.