Let $G$ be a connected real semi-simple Lie group and $H$ a closed connected subgroup. Let $P$ be a minimal parabolic subgroup of $G$. It is shown that $H$ has an open orbit on the flag manifold $G/P$ if and only if it has finitely many orbits on $G/P$. This confirms a conjecture by T. Matsuki.

We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $SU(p,q)$ (resp. $S{O}^{*}\left(2n\right)$, $n$ odd) and the surface group is maximal in some $S\left(U\right(p,p)\times U(q-p\left)\right)\subset SU(p,q)$ (resp. $S{O}^{*}(2n-2)\times SO\left(2\right)\subset S{O}^{*}\left(2n\right)$). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.

In this note, we study formal deformations of derived representations of the principal series representations of $\mathrm{SL}(2,ℝ)$. In particular, we recover all the representations of the derived principal series by deforming one of them. Similar results are also obtained for $\mathrm{SL}(2,ℂ)$.

For the scalar holomorphic discrete series representations of $\mathrm{SU}(2,2)$ and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside $\mathrm{SU}(2,2)$. We construct a Cayley transform between the Ol’shanskiĭ semigroup having $\mathrm{U}(1,1)$ as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for $\mathrm{SU}(2,2)$. This allows calculating the composition series in terms of harmonic analysis on $\mathrm{U}(1,1)$. In particular we show that the Ol’shanskiĭ Hardy space for $\mathrm{U}(1,1)$ is different...