Roger Plymen

The reduced C∗-algebra of the p-adic group GL(n)

Abstract The 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.

A proof of the Baum-Connes conjecture for p-adic GL(n)

Nous donnons une demonstration de la conjecture de Baum-Connes pour le groupe p-adique GL(n).

The Dirac Operator and the Principal Series for Complex Semisimple Lie Groups

The 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.

Periodic cyclic homology of certain nuclear algebras

Abstract Relying on properties of the inductive tensor product, we construct cyclic type homology theories for certain nuclear algebras. In this context, we establish continuity theorems. We compute the periodic cyclic homology of the Schwartz algebra of p-adic GL(n) in terms of compactly supported de Rham cohomology of the tempered dual of GL(n).

C*-algebras and Mackey's axioms

A non-commutative version of probability theory is outlined, based on the concept of aΣ*-algebra of operators (sequentially weakly closedC*-algebra of operators). Using the theory ofΣ*-algebras, we relate theC*-algebra approach to quantum mechanics as developed byKadison with the probabilistic approach to quantum mechanics as axiomatized byMackey. TheΣ*-algebra approach to quantum mechanics includes the case of classical statistical mechanics; this important case is excluded by theW*-algebra approach. By considering theΣ*-algebra, rather than the von Neumann algebra, generated by the givenC*-algebraA in its reduced atomic representation, we show that a difficulty encountered byGuenin concerning the domain of a state can be resolved.

The Hecke algebra of a reductive p-adic group: a geometric conjecture

Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig’s conjecture on the asymptotic Hecke algebra. We prove our conjecture for SL(2) and GL(n). We also prove part (1) of the conjecture for the Iwahori ideals of the groups PGL(n) and SO(5). The conjecture, if true, leads to a parametrization of the smooth dual of G by the points in a complex affine locally algebraic variety.

Base change and K-theory for GL.n/

We investigate base change

Geometric structure in the principal series of the p-adic group G_2

C/R

Chern character for the Schwartz algebra of p-adic GL(n)

at the level of

DISPERSION-FREE NORMAL STATES.

K