1,721,032 research outputs found
The reduced C\sp \ast-algebra of the -adic group
No full textAbstractThe reduced C∗-algebra of the p-adic group GL(n) is Morita equivalent to an abelian C∗-algebra. The structure of this abelian C∗-algebra is described in terms of unramified unitary characters of Levi subgroups. The K-groups K0 and K1 are both free abelian of infinite rank. Generators are essentially parametrized by two items of Langlands data
The Weyl bundle
No full textLet F be a symplectic vector bundle over a space X. We construct a bundle of elementary C*-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F. © 1982
The dirac operator and the principal series for complex semisimple lie groups
No full textThe Dirac operator plays a fundamental role in the geometric construction of the discrete series for semisimple Lie groups. We show that, at the level of K-theory, the Dirac operator also plays a central role in connection with the principal series for complex connected semisimple Lie groups. This proves the Connes-Kasparov conjecture for such groups. © 1983
Geometry of the smooth dual of GL(n)
No full textLet be the smooth dual of the p -adic group G=GL(n). We create on the structure of a complex algebraic variety. There is a morphism of A(n) onto the Bernstein variety ?G which is injective on each component of A(n). The tempered dual of G is a deformation retract of A(n). The periodic cyclic homology of the Hecke algebra of G is isomorphic to the periodised de Rham cohomology supported on finitely many components of A(n). Soit A(n) le dual lisse du groupe p -adique G=GL(n). Nous donnons a A(n) la structure d'une variete algebrique complexe. Il existe un morphisme canonique de A(n) sur la variete de Bernstein G qui est injectif sur chaque composante de A(n). Il y a une retraction par deformation de A(n) sur le dual tempere de G. L'homologie cyclique periodique HP0(H(G)) (resp. HP1(H(G)) ) est isomorphe a la cohomologie de de Rham paire (resp. impaire) a support un nombre fini de composantes du dual lisse de G
Representation theory of -adic groups: a view from operator algebras
No full textOver the past several years, operator algebraists have become increasingly interested in the problem of calculating the K-theory of group C∗-algebras. The focal point of research in this area ist he BaumConnesConjecture[BCH],which proposes a description of K-theory for the C∗-algebra of a group in terms of homology and the representation theory of compact subgroups. Although the main applications of the Baum-Connes Conjecture are to issues in geometry and topology, the conjecture also appears to be of interest from the point of view of harmonic analysis. Whereas for applications to topology one is concerned with discrete groups G (arising as the fundamental groups of manifolds), the conjecture’s links with harmonic analysis appear to be the strongest for reductive Lie groups and p-adic groups. The purpose of these notes is to convey to a reasonably broad audience some byproducts of the authors’ research into the C∗-algebra K-theory of the p-adic group GL(N), which culminated in a proof of the Baum-Connes Conjecture in this case [BHP2]. Along the way to the proof a number of interesting issues came to light which we feel deserve some exposure, even though our understanding of them is far from complete, and is indeed mostly very tentative. Much of what follows is focused on what we call here chamber homology, which is a type of equivariant homology associated to the action of a reductive p-adic group on its Bruhat-Tits affine building. The problem of computing chamber homology can be approached from a number of different directions. An especially interesting problem is to reconcile chamber homology with the Bernstein decomposition for representations of reductive p-adic groups [Be, BD]. This appears to be a far from trivial matter, even in comparatively simple cases. In Section 5 we formulate two very general conjectures which give a broad description of a Bernstein decomposition in chamber homology (perhaps we should call our conjectures questions, since the evidence we have gathered in their favor is not overwhelming).<br/
The representation theory of padic GL(n) and Deligne-Langlands parameters
Get PDFIn this article we cover an episode in the representation theory of GL{n) defined over a p-adic field with finite residue class field. We concentrate on the irreducible tempered representations admitting non-zero Iwahori-fixed vectors. We describe the space of these representations in terms of Deligne
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