Varieties with an action of a reductive group such that the Borel subgroup has an open orbit are called spherical. Spherical varieties are a unifying theme behind many analytic techniques in the theory of automorphic forms, such as the relative trace formula and integral representations of L-functions. After briefly surveying these methods - for which a general and systematic theory is missing - in order to justify this claim, I prove a general result on the representation theory of spherical varieties for split groups over p-adic fields: Irreducible quotients of the "unramified summand" of Cc&#8734(X) (where X is the spherical variety) are "roughly" parametrized by the quotient of a complex torus by a finite reflection group. This generalizes the classical parametrization of the "unramified spectrum" of G by semisimple conjugacy classes in its Langlands dual, and is compatible with recent results of D.Gaitsgory & D.Nadler which assign a "Langlands dual group" to every spherical variety. The main tool in the proof is an action, defined by F.Knop, of the Weyl group of G on the set of Borel orbits on X.

We show that every separable Hilbertian JC*-triple can be decomposed into a countable family of Hilbertian operator spaces, each of which is representable as spaces of creation or annihilation operators. This is
used to give a classification, in the category of operator spaces, of Hilbert spaces which are the range of a contractive projection on a C*-algebra.
(joint work with Matt Neal)