TY - JOUR

T1 - Parabolic induction and restriction via C∗-algebras and Hilbert C∗-modules

AU - Clare, Pierre

AU - Crisp, Tyrone

AU - Higson, Nigel

N1 - Funding Information:
The second author was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). The third author was partially supported by the US National Science Foundation DMS-1101382.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - This paper is about the reduced group C∗-algebras of real reductive groups, and about Hilbert C∗-modules over these C∗-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∗-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∗-algebra to determine the structure of the Hilbert C∗-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.

AB - This paper is about the reduced group C∗-algebras of real reductive groups, and about Hilbert C∗-modules over these C∗-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∗-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∗-algebra to determine the structure of the Hilbert C∗-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.

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U2 - 10.1112/S0010437X15007824

DO - 10.1112/S0010437X15007824

M3 - Article

AN - SCOPUS:84977543866

VL - 152

SP - 1286

EP - 1318

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 6

ER -