Structure of the Unramified L-packet

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