The Langlands Classification and Irreducible Characters for Real Reductive Groups
IIunit(G(R)) :::> IItemp(G(R)) (1. 1) to be the set ofequivalence classes ofirreducible admissible (respectively unitary or tempered) representations of G(R). Now define (G(R)) :::> temp(G(R)) (1. 2) to be the set ofLanglands parameters for irreducible admissible (respec tively tempered) representations of G(R) (see , , ' Chapter 5, and Definition 22. 3). To each ¢ E (G(R)), Langlands attaches a finite set II C II(G(R)), called an L-packet of representations. The L-packets II partition II(G(R)). If ¢ E temp(G(R)), then the representations in II are all tempered, and in this way one gets also a partition of II (G(R)). temp Now the classification of the unitary representations of G(IR) is one of the most interesting unsolved problems in harmonic analysis.
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