Basic functions and unramified local L-factors for split groups

Abstract

According to a program of Braverman, Kazhdan and Ngô Bao Châu, 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.

42 pages, largely revised

Cited by in corpus (1)

• On the formal arc space of a reductive monoid