Basic functions and unramified local L-factors for split groups

TL;DR

This paper studies basic functions related to unramified local L-factors for split groups, deriving their properties, interpreting coefficients via invariant theory, and connecting them to generalized Kostka-Foulkes polynomials.

Contribution

It derives properties of basic functions $f_{\rho,s}$, interprets their coefficients through invariant theory, and links them to generalized Kostka-Foulkes polynomials.

Findings

Basic functions $f_{\rho,s}$ have well-defined properties.

Coefficients of these functions relate to generalized Kostka-Foulkes polynomials.

A rational generating function encodes these coefficients.

Abstract

According to a program of Braverman, Kazhdan and Ng\^o Bao Ch\^au, for a large class of split unramified reductive groups $G$ and representations $\rho$ of the dual group $\hat{G}$, the unramified local $L$-factor $L(s,\pi,\rho)$ can be expressed as the trace of $\pi(f_{\rho,s})$ for a suitable function $f_{\rho,s}$ with non-compact support whenever $\mathrm{Re}(s) \gg 0$. Such functions can be plugged into the trace formula to study certain sums of automorphic $L$-functions. It also fits into the conjectural framework of Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term basic functions; this is supposed to lead to a generalized Tamagawa-Godement-Jacquet theory for $(G,\rho)$. In this article, we derive some basic properties for the basic functions $f_{\rho,s}$ and interpret them via invariant theory. In particular, their coefficients are interpreted as certain generalized Kostka-Foulkes polynomials defined by Panyushev. These coefficients can be encoded into a rational generating function.