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Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group
over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup
is a non-archimedean local field. We establish a bijection between the set of irreducible representations of
and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.
be a (right) Hilbert module over a
is equipped with a left action of a second
, then tensor product with
gives rise to a functor from the category of Hilbert
-modules to the category of Hilbert
-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al. [Parabolic induction and restriction via
-algebras and Hilbert
-modules, Compos. Math.FirstView (2016), 1–33, 2].
This paper is about the reduced group
-algebras of real reductive groups, and about Hilbert
-modules over these
-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
-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
-algebra to determine the structure of the Hilbert
-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.
We present two applications of explicit formulas, due to Cuntz and Krieger, for computations in
-homology of graph
-algebras. We prove that every
-homology class for such an algebra is represented by a Fredholm module having finite-rank commutators, and we exhibit generating Fredholm modules for the
-homology of quantum lens spaces.
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