Basic functions and unramified local L-factors for split groups

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2017,（5）
• 页码范围: 777-812
• 页数: 36页
• 语种: 英文
• DOI: 10.1007/s11425-015-0730-4

摘要

According to a program of Braverman,Kazhdan and Ngo,for a large class of split unramified reductive groups G and representations ρ of the dual group G,the unramified local L-factor L(s,r,ρ) can be expressed as the trace of π(fρ,s) for a function fρ,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,ρ).In this paper,we derive some basic properties for the basic functions fρ,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.