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Parabolic induction and restriction via $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules.
- Source :
-
Compositio Mathematica . Jun2016, Vol. 152 Issue 6, p1286-1318. 33p. - Publication Year :
- 2016
-
Abstract
- This paper is about the reduced group $C^{\ast }$-algebras of real reductive groups, and about Hilbert $C^{\ast }$-modules over these $C^{\ast }$-algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$-algebra to determine the structure of the Hilbert $C^{\ast }$-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0010437X
- Volume :
- 152
- Issue :
- 6
- Database :
- Academic Search Index
- Journal :
- Compositio Mathematica
- Publication Type :
- Academic Journal
- Accession number :
- 116638144
- Full Text :
- https://doi.org/10.1112/S0010437X15007824