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Basic functions and unramified local $$L$$-factors for split groups. (English) Zbl 1437.11077
Summary: According to a program of Braverman, Kazhdan and Ngô, 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 function $$f_{\rho,s}$$ with non-compact support whenever $$\mathrm{Re}(s)\gg0$$. 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 $$(G,\rho)$$. In this paper, 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 D. I. Panyushev [Sel. Math., New Ser. 16, No. 2, 315–342 (2010; Zbl 1248.17006)]. These coefficients can be encoded into a rational generating function.

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields
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