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On some problems suggested by the trace formula. (English) Zbl 0541.22011

Lie group representations II, Proc. Spec. Year, Univ. Md., College Park 1982-83, Lect. Notes Math. 1041, 1-49 (1984).
[For the entire collection see Zbl 0521.00012.]
Some conjectures are formulated concerning representations of reductive algebraic groups. The first section gives some of the local theory. The second section gives the global theory. Conjecture 2.2.1 says that the representations of \(G({\mathbb{A}})\) occuring in the spectral decomposition of \(L^ 2(G(F)\backslash G({\mathbb{A}}))\) occur in packets parameterized by the set of \(L^ 0_ G\) conjugacy classes of maps \[ f:\quad W_ F\times SL(2,{\mathbb{C}})\times SL(2,{\mathbb{C}})\quad \to \quad L_ G. \] Here \(L_ G=LG^ 0\times W_{{\mathbb{R}}}\) is the \(L\)-group of \(G\), and \(W_ F\) is the Weil group. Some examples of \(L\)-groups can be found in the survey article by S. Gelbart [”An elementary introduction to the Langlands program”, Bull. Am. Math. Soc., New Ser. 10, 177-219 (1984)]. The conjecture 2.2.1 further characterizes the discrete spectrum by the finiteness of a certain parameter and then one has a multiplicity formula.
The third section of the paper under review gives a look at the trace formula, which is viewed as motivation for the conjectures. Stabilization of the trace formula is discussed and some examples are considered.
Reviewer: A.Terras

MSC:

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields

Citations:

Zbl 0521.00012