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

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