Let $G = K exp({\mathfrak{a}}^+)K$ be a Cartan decomposition of a connected real linear semisimple Lie group and $m : G \to {\mathfrak{a}}^+$ be the associated map. Let $\Gamma$ be a Zariski dense subgroup of $G$ and $\ell_\Gamma$ be the asymptotic cone to $m(\Gamma)$. This cone is convex and of non empty interior (cf [3]).

We show that $m(\Gamma)$ fills completely $\ell_\Gamma$ in the following sense: for every $\varepsilon \gt 0$ and every closed cone $C$ such that $C - \{0\}$ is included in the interior of $\ell_\Gamma$, every point of $C$ outside a compact is at distance less than $\varepsilon$ from $m(\Gamma)$.

We describe the Howe quotient and theta lift for the trivial representation of $Sp(2n,\mathbb{R})$ for the dual pairs $(O(p,q), Sp(2n,\mathbb{R}))$, within the stable range (i.e., $\min(p, q) \gt 2n$), by explicitly constructing the Howe quotients.

Generalizing the Gelfand hypergeometric system on the Grassmann manifold we introduce certain differential equations on compact Hermitian symmetric spaces and investigate their properties. Especially, we give a criterion for their holonomicity, and show the existence of integral representations of their solutions. We also study properties of certain Radon transforms.

We prove an $S$-arithmetic version, in the context of algebraic groups defined over number fields, of Ratner’s theorem for closures of orbits of subgroups generated by unipotent elements. We apply this result in order to obtain a generalization of results of Margulis and of Borel–Prasad about values of irrational quadratic forms at integral points to the general setting of hermitian forms over division algebras with involutions of first or second kind. As a byproduct of our considerations we obtain another proof of the strong approximation theorem for algebraic groups defined over number fields.

The Langlands classification theorem describes all admissible representations of a reductive group $G$ in terms of the tempered representations of Levi subgroups of $G$. I will describe work with Susana Salamanca-Riba that provides (conjecturally) a similar description of the unitary representations of $G$ in terms of certain very special unitary representations of Levi subgroups.

In this note we extend the Kac and Todorov's orbifold theory to the principal admissible representations. This gives rise to nonsymmetric transformation matrices.