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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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