## Séminaire Orléans

**Invariant integral operators on the Oshima compactification of a Riemannian symmetric space**

Aprameyan Parthasarathy (Marburg)

Thursday 01 December 2011 15:00 - Orléans - Salle PTICREM

**Résumé :**

Let $\X\simeq G/K$ be a Riemannian symmetric space of non-compact type, $\widetilde\X$ its Oshima compactification, and $(\pi,\mathrm{C}(\widetilde\X))$ the regular representation of $G$ on $\widetilde\X$. In this talk, we study integral operators on $\widetilde\X$ of the form $\pi(f)$, where $f$ is a rapidly falling function on $G$, and characterize them within the framework of pseudodifferential operators, describing the singular nature of their kernels. In particular, we consider the holomorphic semigroup generated by a strongly elliptic operator associated to the representation $\pi$, as well as its resolvent, and describe the asymptotic behavior of the corresponding semigroup and resolvent kernels. We then define a regularized trace for the convolution operators $\pi(f)$, yielding a distribution on $G$ which can be interpreted as a global character of $\pi$. In case that $f$ has compact support in a certain open set of transversal elements, this distribution is a locally integrable function, and is given by a fixed point formula analogous to the formula for the global character of an induced representation of $G$.

**Liens :**