Structure of the Unramified L-packet

TL;DR

This paper investigates the structure of unramified L-packets for connected reductive groups over local fields, revealing how their parametrization relates to conjugacy classes of hyperspecial subgroups and the action of a finite abelian group.

Contribution

It establishes a natural homogeneous space structure for the L-packet parameter group and describes the relationship between hyperspecial subgroup conjugacy classes and L-packet representations.

Findings

The L-packet parameter group is a homogeneous space for a finite abelian group.

Conjugacy classes of hyperspecial subgroups form a principal homogeneous space for a finite abelian group.

The non-vanishing of representations is preserved under the action of the abelian group.

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 $\lambda$ 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 $\hat{\Omega}$. Also, the $L$-packet $\Pi(\varphi_{\lambda})$ 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 $\pi_{\rho}\in\Pi(\varphi_{\lambda})$, where $\rho\in\hat{R}$ and let $[K]$ denote the conjugacy class of hyperspecial subgroup $K.$ Then we show that $\pi_{\rho}^{K}\neq0$ if and only if $\pi_{\omega\cdot\rho}^{K_{\omega}}\neq0$ where $\omega\in\hat{\Omega}$ and $K_{\omega}$ is any hyperspecial subgroup in the conjugacy class $\omega\cdot[K]$.