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Parameters for twisted representations. (English) Zbl 1342.22021
Nevins, Monica (ed.) et al., Representations of reductive groups. In honor of the 60th birthday of David A. Vogan, Jr. Proceedings of the conference, MIT, Cambridge, MA, USA, May 19–23, 2014. Cham: Birkhäuser/Springer (ISBN 978-3-319-23442-7/hbk; 978-3-319-23443-4/ebook). Progress in Mathematics 312, 51-116 (2015).
In the paper [the authors et al., “Unitary representations of real reductive groups”, Preprint, arxiv:1212.2192], an algorithm was constructed for the calculation of a signature of the Hermitian form on an irreducible representation of a real reductive Lie group. “The starting point for the algorithm is the Langlands classification which provides a parameter space for the irreducible unitary representations of \(G\). In order to determine the unitary representations it is necessary to pass to a large extended group \(G^{\delta}\) containing \(G\) of index \(2\) and to construct a parameter space for the representations of \(G^{\delta}\). The pupose of this paper is to address the following problem: when a parameter for \(G\) extends to \(G^{\delta}\) in two ways there is no canonical way to chose one of the extensions. Consequently the theory for \(G\) does not carry over to \(G^{\delta}\) in a simple way and it is necessary to define parameters for \(G^{\delta}\) and to study their properties in some detail”.
For the entire collection see [Zbl 1336.22001].

22E46 Semisimple Lie groups and their representations
20G05 Representation theory for linear algebraic groups
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