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Basic functions and unramified local L-factors for split groups

According to a program of Braverman, Kazhdan and Ngô Bao Châu, for a large class of split unramified reductive groups $G$ and representations $ρ$ of the dual group $\hat{G}$, the unramified local $L$-factor $L(s,π,ρ)$ can be expressed as the trace of $π(f_{ρ,s})$ for a suitable function $f_{ρ,s}$ 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 $(G,ρ)$. In this article, we derive some basic properties for the basic functions $f_{ρ,s}$ 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.