Branching rule for classical Lie algebras in positive characteristic

Question

The restriction of an irreducible $\mathfrak{sl}_n(\mathbb{C})$-module to $\mathfrak{sl}_{n-1}(\mathbb{C})$ is described by a branching rule which says that if $L(\lambda)$ is the simple $\mathfrak{sl}_n(\mathbb{C})$-module corresponding to a partition $\lambda$ then its restriction is a direct sum of $L(\mu)$s where the $\mu$ are obtained by removing some boxes from $\lambda$.

Now consider the analogous problem for $\mathfrak{sl}_n(k)$, where $k$ is a field of characteristic $p>0$. It is no longer generally true that simple modules restrict to semisimple modules. My question is whether a branching rule determining the indecomposable summands of the restriction of a simple module is known. A general answer may be too much to ask, but I would be interested in references for nontrivial special cases, or for small $n$, or for other classical Lie algebras.

Answer 1

As my comment indicated, there is currently little hope of writing down general branching rules in characteristic $p$. In fact, given the history of work on Lusztig's conjecture about formal characters of simple modules, it's unclear whether good "formulas" as such will exist in this subject; perhaps one has to settle for recursive calculations, which may or may not be feasible except in small cases.

In a more definite direction, it's useful here to switch from Lie algebras to algebraic groups: the Lie algebra irredudibles with "restricted" highest weights relative to $p$ yield all irreducibles for the group as twisted tensor products (Steinberg). Here the foundations are well organized in Jantzen's book. For special linear groups as for others, only in small ranks or for small $p$ can one determine explicitly the irreducibles. The Weyl modules have the advantage of giving known weight combinatorics, but the disadvantage of usually being far from irreducible.

It's also wise, even for special linear groups, to think in terms of restriction to arbitrary Levi factors of parabolic subgroups. This includes a special linear group of rank one less than the given group, but is more flexible as a framework for branching.

There isn't a lot of explicit study of such questions, I think, but interesting work in special cases has been done by people associated (now or formerly) with Minsk: A.S. Kleshchev, I.D. Suprunenko, and others. An example is the recent work of Anna Osinovskaya, On the restrictions of modular irreducible representations of algebraic groups of type $A_n$ to naturally embedded subgroups of type $A_2$. J. Group Theory 8 (2005), no. 1, 43–92.

Earlier work by Kleshchev dealt with general linear and finite symmetric groups in the spirit of Schur-Weyl duality. Here too there are some partial insights into branching problems: A remark on modular branching rules for reductive groups. Mat. Zametki 56 (1994), no. 5, 143--145; translation in Math. Notes 56 (1994), no. 5-6, 1195–1196 (1995). Some work by Steve Donkin has also dealt with the restriction problem, sometimes in a different context. For instance, he has introduced systematic use of indecomposable "tilting" modules for algebraic groups; these restrict nicely to Levi subgroups: On tilting modules for algebraic groups. Math. Z. 212 (1993), no. 1, 39–60. (But the details about tilting modules are complicated and mostly unknown.)

Probably the best advice in this area of representation theory is to formulate questions narrowly and cautiously, for instance about irreducibles having only one Weyl group orbit of weights.