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

Let $G$ be a real reductive group. Let $\pi_1$ and $\pi_2$ be unitary irreducible representations of $G$. The decomposition of the tensor product $pi_1\otimes\pi_2$ has been a long-standing problem in harmonic analysis. In this talk, we will discuss this problem for the case where $G=Spin(n, 1)$. It turns out that the decomposition of $pi_1\otimes\pi_2$ in this case is closely related to the branching problem of unitary irreducible representations of $G$ with respect to a minimal parabolic subgroup $P$. Especially, in the case where $pi_1$ is a unitary principal series (and $pi_2$ is an arbitrary unitary irreducible representation of $G$), the tensor product $pi_1\otimes\pi_2$ can be decomposed explicitly based on the knowledge of explicit branching laws with respect to $P$ and other results and techniques in harmonic analysis and representation theory. If time permits, we will also discuss the case where $pi_1$ is a complementary series. This is ongoing joint work with S. Afentoulidis-Almpanis.