The Selberg-Arthur trace formula. Based on lectures by James Arthur
type: Item
instance of: scholarly article
title: The Selberg-Arthur trace formula. Based on lectures by James Arthur
zbMATH Open document ID: 0769.11026
author: Salahoddin Shokranian
published in: Lecture Notes in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: There are at least two approaches to the trace formula. Approach \(A\) involves automorphic representations of adele groups. Approach \(C\) involves the spectral theory of the Laplace and other \(G\)-invariant differential operators on a symmetric space \(G/K\) and the classical theory of automorphic forms. The present monograph is clearly type \(A\). It comes from 1987 lectures of J. Arthur. Let's consider a few examples. Example 1. Finite Groups [from p. 18 of the monograph under review]. Suppose that \(G\) is a finite group. Then a (finite-dimensional) unitary representation \(\pi\) of \(G\) is a group homomorphism \(\pi: G\to U(d_ \pi)\). Here \(U(n)\) is the unitary group of \(n\times n\) complex matrices \(g\) such that \(g^* g=I\), where \(g^*\) denotes the transpose conjugate of \(g\). The unitary representation \(\pi\) is irreducible if it is not uniformly block diagonalizable. Two representations are equivalent if one can be obtained from the other by uniform change of basis. Let \(\widehat G\) be a complete set of inequivalent irreducible unitary representations of \(G\). If \(f: G\to\mathbb{C}\), then the Fourier transform is \(\widehat f(\pi)=\sum_{g\in G} f(g) \pi(g)\). Note that \(\widehat f(\pi)\) is a \(d_ \pi\times d_ \pi\) matrix. Next let's discuss the Selberg trace formula for a finite group \(G\) with subgroup \(H\). Suppose for simplicity that \(\omega\) is a one-dimensional representation of \(H\). Let \(\rho\) be the induced representation of \(G\); i.e. \(\rho=\text{Ind}^ G_ H \omega\), acting on the space \[ V_ \omega=\bigl\{\phi: G\to\mathbb{C}\mid\phi(hx)=\omega(h)\phi(x),\;\text{for all }x\in G,\;h\in H\bigr\} \] by the right action \(\rho(g)\phi(x)=\phi(xg)\), for \(x,g\in G\). We can extend the definition of the Fourier transform above to arbitrary representations, not just irreducible ones. Then we find that for \(\phi\in V_ \omega\) we have \[ \bigl[\widehat f(\rho)\phi\bigr](x)=\sum_{y\in G} f(y) \phi(xy)=\sum_{y\in H\backslash G}\…
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 53220
zbMATH Keywords: Arthur's trace formula for \(GL(n)\)
zbMATH Keywords: adelic weighted orbital integrals
zbMATH Keywords: automorphic representations
zbMATH Keywords: adele groups
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: finite group
zbMATH Keywords: Maass wave form
zbMATH Keywords: Roelcke-Selberg spectral decomposition
zbMATH Keywords: Arthur's reduction theory
zbMATH Keywords: kernel functions
zbMATH Keywords: Langlands-Arthur formula
zbMATH Keywords: inner products of truncated Eisenstein series
reviewed by: Audrey A. Terras
MaRDI profile type: Publication
DOI: 10.1007/BFB0092305
description: scientific article; zbMATH DE number 53220