Decomposition of Tensor Products Involving a Steinberg Module

Abstract

We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G r or a finite Chevalley group G(F q ). In all three cases, we give formulas reducing this to standard character data for G.Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G r or G(F q ). Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin's tilting conjecture.

On (𝑝,𝑟)-filtrations and tilting modules

“…In this section we look at how various tilting modules decompose over G and G r . Since St r ⊗L(λ), where λ ∈ X r (T ), is known to be tilting in many cases, these results immediately apply to such modules (see also [9] for decompositions of modules of the form St r ⊗ ∇(λ)).…”

A Conjectural Algorithm for Simple Characters of Algebraic Groups

“…• M, N is the Euler characteristic of the modules M and N , see [Kil17] for more details as well as an alternative definition.…”

“…We will make extensive use of the form denoted by •, • in [Kil17]. Apart from the properties of this form we discuss as Conjecture 4.1, the main properties we will need are:…”