A Hecke Algebra Approach to p-adic Functionals

Abstract

Unique model spaces for representations of reductive groups over $p$-adic fields play an integral role in the theory of automorphic forms (where for our purposes, 'unique' means the decomposition of the model space is multiplicity-free). Uniqueness facilitates precise computation of special functions in the model as in the work of Casselman, Shalika, and Shintani \cite{casselman, casselman-shalika, shintani}, and is a common feature of local components of global integral representations of $L$-functions, as in Godement and Jacquet \cite{godement-jacquet}, and Ginzburg and Rallis \cite{ginzburg-rallis}. Here we study uniqueness of local model spaces with respect to 'universal unramified principal series' as outlined in Haines, Kottwitz, and Prasad \cite{HKP}. In this program, the convolution algebra of compactly supported, Iwahori-biinvariant functions on $G(\Q_p)$ (the \emph{Iwahori-Hecke algebra}, henceforth denoted as $\mc{H}$), provides powerful algebraic structure to the theory of $p$-adic group representations and allows one to simultaneously study the full unramified principal series. A recurring theme is that unique models and unique functionals on unramified principal series representations are associated to Hecke algebra modules of the form $\ind_{\mc{H}_0}^\mc{H} \varepsilon$, where $\mc{H}_0$ is the finite Hecke algebra consisting of functions in $\mc{H}$ supported on the integer points in $G(\Q_p)$ and $\varepsilon$ is a linear character of $\mc{H}_0$. Brubaker, Bump, and Friedberg \cite{BBF} show that many standard unique functionals map to \emph{left} induced $\mc{H}$-modules of this form, and Chan and Savin \cite{chan-savin-bessel, chan-savin-iwahori} show that the Iwahori-fixed vectors in certain standard unique model spaces are associated to \emph{right} $\mc{H}$-modules of this form.\\ We explore and expand this program in several ways. We provide sufficient conditions for an $\mc{H}$-module to be of the form $\ind_{\mc{H}_0}^\mc{H} \varepsilon$, expanding the $GL_n$ case described in \cite{CS-BZ}. We show that \emph{left} $\mc{H}$-modules on the functional studied by Brubaker, Bump, and Friedberg \cite{BBF} are essentially the same as the \emph{right} $\mc{H}$-modules on the model side identified by Chan and Savin \cite{chan-savin-bessel, chan-savin-iwahori}. We then classify, under certain conditions, the $\mc{H}$-modules which are associated to either unique modules or unique functionals. Finally, we investigate possible generalizations of this theory to finite multiplicity (but not unique) model spaces, specifically the generalized Gelfand-Graev representations of Kawanaka \cite{KawI, KawS} (\emph{generalized Whittaker models} in the program of Moeglin and Waldspurger \cite{MW}) of both $G(\F_q)$ and $G(\Q_p)$.