Appears in collection : 2025 - T1 - WS1 - Intertwining operators and geometry

One major question in the representation theory of locally compact groups is how an irreducible representation of a group $G$ decomposes if restricted to a subgroup $H$. For $\pi$ and $\tau$ irreducible representations of $G$ and $H$, respectively, elements of $Hom_H(\pi\vert_{H}, \tau)$ are referred to as symmetry breaking operators, a term coined by Kobayashi. In a recent joint paper with Jan Frahm we initiate the study of symmetry breaking operators over the $p$-adic fields. More precisely, we consider the pair $(PGL_2(E), PGL_2(F))$, when $E$ is a quadratic field extension of an arbitrary $p$-adic field $F$, and explicitly construct and provide a classification of all symmetry breaking operators between principal series representations of $PGL_2(E)$ and $PGL_2(F))$. Although our results are very similar to the Archimedean case, this talk will try to focus not only on the similarities but also on the differences between the Archimedean and non-Archimedean situations.