A Gelfand-Graev Formula and Stable Transfer Factors in the Unramidied Case for $\text{SL}_\ell(F)$ and $\text{GL}_\ell(F)$, $\ell$ an odd Prime

Abstract: Let $F$ be a nonarchimedean local field of characteristic 0 with residual characteristic $p$ and let $\ell$ be an odd prime with $2\ell<p$. We establish and explicitly compute the local stable transfer factor $\Theta_\phi$ in the sense of \cite{SetT} associated to a natural $L$-embedding $\phi:{^LT}\to{^LG}$ for $G=\text{SL}\ell$ for $\ell$ an odd prime and $T\subset G$ a maximal unramified elliptic torus defined over $F$. We also explicitly compute the associated stable transfer, answering in the affirmative the Questions A and B of \cite{SetT}. We do the same, explicitly computing the stable transfer factor $\Theta{\widetilde{\phi}}$ and the associated stable transfer operator, in the related case of $\widetilde{\phi}:{^L\widetilde{T}}\to{^L\widetilde{G}}$ for $\widetilde{G}=\text{GL}_\ell$ and $\widetilde{T}\subset \widetilde{G}$ a maximal unramified elliptic torus defined over $F$.