On the residual spectrum of GL(n).

*(English)*Zbl 0539.22016
Lie group representations II, Proc. Spec. Year, Univ. Md., College Park 1982-83, Lect. Notes Math. 1041, 185-208 (1984).

[For the entire collection see Zbl 0521.00012.]

Let F be a number field, G be GL(n) over F, Z the center of G, A the adeles of F, w a character of Z(A)/Z(F). Let \(L^ 2_ d(w)\) be the discrete part of the Hilbert space of functions on G(A)/G(F) which transform under the center by w and which are square-integrable modulo the center. G(A) acts on it and it has a Hilbert direct-sum decomposition into irreducible representations. Extending results of B. Speh, the author constructs some of the irreducible components B(s,v,d) of \(L^ 2_ d(w)\) using residues of Eisenstein series. These are parametrized by a divisor d of n, a character v such that \(v^ d=w\), and a cuspidal automorphic representation s of GL(r,A) with central character v (and \(r=n/d)\). The corresponding Eisenstein series is built on the parabolic subgroup of G of type (r,r,...,r).

The author ”would like to prove” that \(L^ 2_ d(w)=\oplus B(s,v,d)\). Even the multiplicity of a single B(s,v,d) is an open problem, as well as the question of exhaustion. The bulk of this paper goes to prove a partial result: set \(d=n\), \(w=s=v=1\) (trivial character), \(F={\mathbb{Q}}\), K a maximal compact subgroup of G(A). Then the K-invariant functions in \(L^ 2_ d(1)\) consist only of B(1,1,n), i.e. only of constant functions. The methods of proof include Langlands’ theory of Eisenstein series and computation of derivatives of ”residual Eisenstein series”. The problem boils down to questions on root systems. Most of the proofs are only sketched.

Let F be a number field, G be GL(n) over F, Z the center of G, A the adeles of F, w a character of Z(A)/Z(F). Let \(L^ 2_ d(w)\) be the discrete part of the Hilbert space of functions on G(A)/G(F) which transform under the center by w and which are square-integrable modulo the center. G(A) acts on it and it has a Hilbert direct-sum decomposition into irreducible representations. Extending results of B. Speh, the author constructs some of the irreducible components B(s,v,d) of \(L^ 2_ d(w)\) using residues of Eisenstein series. These are parametrized by a divisor d of n, a character v such that \(v^ d=w\), and a cuspidal automorphic representation s of GL(r,A) with central character v (and \(r=n/d)\). The corresponding Eisenstein series is built on the parabolic subgroup of G of type (r,r,...,r).

The author ”would like to prove” that \(L^ 2_ d(w)=\oplus B(s,v,d)\). Even the multiplicity of a single B(s,v,d) is an open problem, as well as the question of exhaustion. The bulk of this paper goes to prove a partial result: set \(d=n\), \(w=s=v=1\) (trivial character), \(F={\mathbb{Q}}\), K a maximal compact subgroup of G(A). Then the K-invariant functions in \(L^ 2_ d(1)\) consist only of B(1,1,n), i.e. only of constant functions. The methods of proof include Langlands’ theory of Eisenstein series and computation of derivatives of ”residual Eisenstein series”. The problem boils down to questions on root systems. Most of the proofs are only sketched.

Reviewer: A.Ash

##### MSC:

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |