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The unitary dual of the universal covering group of \(\mathrm{GL}(n,\mathbb{R})\). (English) Zbl 0732.22010
The author classifies the irreducible unitary representations of the double cover \(G\) of \(\mathrm{GL}(n,\mathbb{R})\) which do not factor to \(\mathrm{GL}(n,\mathbb{R})\) itself. The methods and techniques are similar to D. Vogan’s in his classification of the unitary dual of \(\mathrm{GL}(n,\mathbb{R})\) [Invent. Math. 83, 449–505 (1986; Zbl 0598.22008)]. One finds that just one new building block appears in the unitary dual of \(G\). This is a unipotent representation whose Langlands parameters are obtained from those of a well-known unitary representation of \(\mathrm{GL}(n,\mathbb{R})\) by dividing by two. All unitary representations are then obtained from one of the building blocks by unitary induction and Stein complementary series.

MSC:
22E46 Semisimple Lie groups and their representations
20G05 Representation theory for linear algebraic groups
22D10 Unitary representations of locally compact groups
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