The Non-Split Bessel Model on GSp(2n) as an Iwahori-Hecke Algebra Module

TL;DR

This paper constructs and analyzes a non-split Bessel model for GSp(2n), connecting it to Iwahori-Hecke algebra modules, and extends known results from GSp(4) to higher dimensions under a conjecture.

Contribution

It explicitly realizes the non-split Bessel model as a Hecke algebra module for GSp(2n) and extends existence and uniqueness results to higher dimensions assuming a conjecture.

Findings

Explicit formula for the spherical vector in the Bessel model

Connection between Iwahori-fixed vectors and Hecke algebra characters

Extension of results to GSp(2n) under a conjecture

Abstract

We realize the non-split Bessel model of Novodvorsky and Piatetski-Shapiro as a generalized Gelfand-Graev representation of GSp(4), as suggested by Kawanaka. With uniqueness of the model already established by Novodvorsky and Piatetski-Shapiro, we establish existence of a Bessel model for unramified principal series representations. We then connect the Iwahori-fixed vectors in the Bessel model to a linear character of the Hecke algebra of GSp(4) following the method outlined more generally by Brubaker, Bump, and Friedberg. We use this connection to calculate the image of Iwahori-fixed vectors of unramified principal series in the model, and ultimately provide an explicit alternator expression for the spherical vector in the model. We show that the resulting alternator expression matches previous results of Bump, Friedberg, and Furusawa. We offer the conjecture that a generalized Bessel model on GSp(2n) retains the uniqueness property estabilished in the case when $n=2$; assuming that this conjecture holds, we extend all of the previously mentioned results to the case where $n>2$, including existence of the model for unramified principal series representations.