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Register a new user$C^ast$-blocks and crossed products for classical $p$-adic groups
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Let $G$ be a real or $p$-adic reductive group. We consider the tempered dual of $G$, and its connected components. For real groups, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component has a simple geometric structure which encodes the reducibility of induced representations. For $p$-adic groups, each connected component of the tempered dual comes with a compact torus equipped with a finite group action, and we prove that a version of Wassermanns theorem holds true under a certain geometric assumption on the structure of stabilizers for that action. We then focus on the case where $G$ is a quasi-split symplectic, orthogonal or unitary group, and explicitly determine the connected components for which the geometric assumption is satisfied.
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A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations $pi colon G to U(cH)$. In this paper we present a new approach to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators whose range is contained in the space $cH^infty$ of smooth vectors. Our first major result is a characterization of smoothing operators $A$ that in particular implies smoothness of the maps $pi^A colon G to B(cH), g mapsto pi(g)A$. The concept of a smoothing operator is particularly powerful for representations $(pi,cH)$ which are semibounded, i.e., there exists an element $x_0 ing$ for which all operators $iddpi(x)$, $x in g$, from the derived representation are uniformly bounded from above in some neighborhood of $x_0$. Our second main result asserts that this implies that $cH^infty$ coincides with the space of smooth vectors for the one-parameter group $pi_{x_0}(t) = pi(exp tx_0)$. We then show that natural types of smoothing operators can be used to obtain host algebras and that, for every metrizable Lie group, the class of semibounded representations can be covered completely by host algebras. In particular, it permits direct integral decompositions.
We present a geometric proof of Bernsteins second adjointness for a reductive $p$-adic group. Our approach is based on geometry of the wonderful compactification and related varieties. Considering asymptotic behavior of a function on the group in a neighborhood of a boundary stratum of the compactification, we get a co-specialization map between spaces of functions on various varieties with $Gtimes G$ action. These maps can be viewed as maps of bimodules for the Hecke algebra, and the corresponding natural transformations of functors lead to the second adjointness. We also get a formula for the co-specialization map expressing it as a composition of the orishperic transform and inverse intertwining operator; a parallel result for $D$-modules was obtained in arXiv:0902.1493. As a byproduct we obtain a formula for the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra, generalizing a result by Opdam.
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 the sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction.
We calculate the Plancherel formula for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. As a consequence we obtain a concrete description of their associated reduced group $ C^* $-algebras. The main ingredients in our proof are the Bernstein-Gelfand-Gelfand complex and the Hopf trace formula.
In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthurs main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogans work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.
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