Principal series component of Gelfand-Graev representation

Abstract

Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. Let $B$ be a minimal $F$-parabolic subgroup with Levi factor $T$ and unipotent radical $U$. Let $\psi$ be a non-degenerate character of $U(F)$ and $\lambda$ a character of $T(F)$. Let $(K,\rho )$ be a Bushnell-Kutzko type associated to the Bernstein block of $G(F)$ determined by the pair $(T,\lambda )$. We study the $\rho$-isotypical component $(c{\text{-ind}}_{U(F)}^{G(F)}\psi {)}^{\rho }$ of the induced space $c{\text{-ind}}_{U(F)}^{G(F)}\psi$ of functions compactly supported mod $U(F)$. We show that $(c{\text{-ind}}_{U(F)}^{G(F)}\psi {)}^{\rho }$ is cyclic module for the Hecke algebra $\mathcal{H}(G,\rho )$ associated to the pair $(K,\rho )$. When $T$ is split, we describe it more explicitly in terms of $\mathcal{H}(G,\rho )$. We make assumptions on the residue characteristic of $F$ and later also on the characteristic of $F$ and the center of $G$ depending on the pair $(T,\lambda )$. Our results generalize the main result of Chan and Savin in \cite{CS18} who treated the case of $\lambda =1$ for $T$ split.

Statement of Theorem 3 corrected. To appear in Proceedings of the American Mathematical Society

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