\(L\)-groups, projective representations, and the Langlands classification. (English) Zbl 0760.22013

Let \(G\) be a reductive algebraic group defined over the reals and \(G(\mathbb{R})\) its group of real points. The dual group \(^ L G^ 0\) of \(G\) is a complex reductive group whose roots are the coroots of \(G\). In his paper [in Representation theory and harmonic analysis on semisimple Lie groups, Math. Surv. Monogr. 31, 101-170 (1989; Zbl 0741.22009)], R. P. Langlands introduced the \(L\)-group \(^ L G\) of \(G\) as a certain semidirect product of the dual group with the Galois group of \(\mathbb{C}\) over \(\mathbb{R}\). He also gave a parametrization of the irreducible admissible representations of \(G(\mathbb{R})\) in terms of the \(L\)-group. In this paper the authors rework and expand some of Langlands’ ideas. They introduce a larger class of extensions of the Galois group by the dual group. They call these \(E\)-groups and prove that they are related to certain projective representations of \(G(\mathbb{R})\).


22E46 Semisimple Lie groups and their representations
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
11R39 Langlands-Weil conjectures, nonabelian class field theory


Zbl 0741.22009
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