Structure of the Unramified L-packet

Abstract

Let $\boldsymbol{G}$ be an unramified connected reductive group defined over a non-archemedian local field $k$ and let $\boldsymbol{T}$ be a maximal torus in $\boldsymbol{G}.$ Let $λ$ be an unramified character of $\boldsymbol{T}.$ Then the conjugacy classes of hyperspecial subgroups of $\boldsymbol{G}(k)$ is a principal homogenous space for a certain finite abelian group $\hatΩ$. Also, the $L$-packet $Π(φ_λ)$ associated to $λ$ is parametrized by an abelian group $\hat{R}$. We show that $\hat{R}$ is naturally a homogenous space for $\hatΩ$. Further, let $π_ρ\inΠ(φ_λ)$, where $ρ\in\hat{R}$ and let $[K]$ denote the conjugacy class of hyperspecial subgroup $K.$ Then we show that $π_ρ^{K}\neq0$ if and only if $π_{ω\cdotρ}^{K_ω}\neq0$ where $ω\in\hatΩ$ and $K_ω$ is any hyperspecial subgroup in the conjugacy class $ω\cdot[K]$.