Shalika Models for Finite General Linear and Unitary Groups

Abstract

We completely and explicitly classify the irreducible representations of finite general linear groups ${\mathrm{GL}}_{2\mathfrak{n}}({\mathbb{F}}_q)$ and finite unitary groups ${\mathrm{U}}_{2\mathfrak{n}}({\mathbb{F}}_q)$ that admit a Shalika model. Using Lusztig's Jordan decomposition of irreducible representations, we establish a multiplicity formula for Shalika multiplicities in terms of the Weyl group of the centralizer of a semisimple element in the dual group. We also derive connections to symplectic models, linear models, and relative Langlands duality.