Structure of the Unramified L-packet

Abstract

Let $\mathbf{G}$ be an unramified connected reductive group defined over a non-archemedian local field $k$ and let $\mathbf{T}$ be a maximal torus in $\mathbf{G}.$ Let $\lambda$ be an unramified character of $\mathbf{T}.$ Then the conjugacy classes of hyperspecial subgroups of $\mathbf{G}(k)$ is a principal homogenous space for a certain finite abelian group $\hat{\Omega }$. Also, the $L$-packet $\Pi ({\phi }_{\lambda })$ associated to $\lambda$ is parametrized by an abelian group $\hat{R}$. We show that $\hat{R}$ is naturally a homogenous space for $\hat{\Omega }$. Further, let ${\pi }_{\rho }\in \Pi ({\phi }_{\lambda })$, where $\rho \in \hat{R}$ and let $[K]$ denote the conjugacy class of hyperspecial subgroup $K.$ Then we show that ${\pi }_{\rho }^K\ne 0$ if and only if ${\pi }_{\omega \cdot \rho }^{K_{\omega }}\ne 0$ where $\omega \in \hat{\Omega }$ and $K_{\omega }$ is any hyperspecial subgroup in the conjugacy class $\omega \cdot [K]$.

29 Pages. Minor organizational changes and corrections. Typos fixed

References in corpus (1)

• Arthur Packets and the Ramanujan Conjecture