According to a program of Braverman, Kazhdan and Ngo, for a large class of split unramified reductive groups G and representations p of the dual group G, the unramified local L-factor L(s, π, p) can be expressed as the trace of π(f_p,s) for a function f_p,s with non-compact support whenever Re(s)》0. Such a function should have useful interpretations in terms of geometry or combinatorics, and it 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, p). In this paper, we derive some basic properties for the basic functions f_p,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.
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