Tate cohomology of unipotent cuspidal representations of finite groups of Lie type

Abstract

Let $p$ be a prime and $k$ be an algebraic closure of the finite field with $p$ elements. Let $G$ be a connected reductive algebraic group over $k$, and let $F:G\rightarrow G$ be a Frobenius morphism, defining an $\mathbb{F}_q$-rational structure on $G$, where $q$ is a power of $p$. Let $\ell$ be a prime different from $p$ and $\ell\ne 2$. Let $\Pi$ be an $\ell$-adic cuspidal unipotent representation of $G^{F^\ell}$ such that $\Pi\simeq \Pi^\sigma$, where $\sigma$ is a generator the Galois group ${\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$. In this paper, we prove the irreducibility of the zeroth Tate cohomology group $\widehat{\rm H}^0(\sigma,\mathcal{L})$ and vanishing of the first Tate cohomology group $\widehat{\rm H}^1(\sigma,\mathcal{L})$ for any $G^{F^\ell}\rtimes{\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$-stable lattice $\mathcal{L}$ in $\Pi$, using the multiplicity-one property of a progenerator constructed via the generalized Gelfand-Graev representations and Harish-Chandra induction.