Let $p$ be prime. In the representation theory of $p$-adic groups over algebraically closed fields of characteristic $p$, such as the general linear group $GL_n(F)$ or the special linear group $SL_n(F)$, where the entries vary in a finite extension $F$ of the field $\mathbb{Q}_p$ of $p$-adic numbers, an important concept is that of distinguished representations. Given a subgroup $H$ of $G$ and a character $\chi$ of $H$ (i.e. a group homomorphism from $H$ to $\mathbb{K}^*$), a representation $(\pi, V)$ of $G$ is said to be $(H, \chi)$-distinguished when

\[ Hom_H (\pi, \chi) := \{l: \, V \to \chi \, \text{linear map} : \, \forall h \in H, \, v \in V, l(\pi(h)v) = \chi(h)l(v) \} \ne 0.\]

When $\chi$ is trivial (i.e. $\chi= \mathbb{1}$), we simply say that $(\pi, V)$ is $H$-distinguished. We study this concept of $p$-modular representation of $SL_2(\mathbb{F}_q)$, focusing in this talk on distinction with respect to its standard Borel subgroup of upper-triangular matrices.

This is joint work with Ramla Abdellatif (Université de Picardie Jules Verne) and Jocelyn P. Vilela (Mindanao State University-IIT).

In this talk, I would like to present some recent results on the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the stationary nonlinear Schrödinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the $L^2$-sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this talk, we consider the problem in bounded domains, in the presence of weakly attractive potentials, or under trapping potentials, and investigate the following questions:

(i) whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and

(ii) if the existence of a mountain-pass solution persists.

We provide positive answers under suitable assumptions. This is based on joint work with Dario Pierotti and Gianmaria Verzini (Politecnico di Milano).