The Weil-Steinberg character of finite classical groups

TL;DR

This paper analyzes the product of Weil and Steinberg characters in finite classical groups, revealing multiplicity properties and applications to subgroup restrictions, with new insights into modular representations and indecomposable summands.

Contribution

It computes irreducible constituents of Weil and Steinberg character products in classical groups, showing multiplicity freeness in certain cases and exploring modular representation aspects.

Findings

Product is multiplicity free for symplectic and unitary groups

Product is not multiplicity free for general linear groups

Restriction of Steinberg character to stabilizer subgroup is multiplicity free

Abstract

We compute the irreducible constitutents of the product of the Weil character and the Steinberg character in those finite classical groups for which a Weil character is defined, namely the symplectic, unitary and general linear groups. It turns out that this product is multiplicity free for the symplectic and general unitary groups, but not for the general linear groups. As an application we show that the restriction of the Steinberg character of such a group to the subgroup stabilizing a vector in the natural module is multiplicity free. The proof of this result for the unitary groups uses an observation of Brunat, published as an appendix to our paper. As our "Weil character" for the symplectic groups in even characteristic we use the 2-modular Brauer character of the generalized spinor representation. Its product with the Steinberg character is the Brauer character of a projective module. We also determine its indecomposable direct summands.