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 ${\Theta }_{\varphi }$ in the sense of \cite{SetT} associated to a natural $L$-embedding $\varphi :{}^{L}T\to {}^{L}G$ 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 ${\Theta }_{\tilde{\varphi }}$ and the associated stable transfer operator, in the related case of $\tilde{\varphi }:{}^L\tilde{T}\to {}^L\tilde{G}$ for $\tilde{G}={\text{GL}}_{\ell }$ and $\tilde{T}\subset \tilde{G}$ a maximal unramified elliptic torus defined over $F$.

References in corpus (2)

• Explicit asymptotic expansions for tame supercuspidal characters
• On the Steinberg character of a semisimple p-adic group