Intertwining operators and geometry

Tensor product decomposition for rank one spin groups

21 janv. 2025, 11:00

50m

Amphithéâtre Hermite (Institut Henri Poincaré)

Orateur

Gang Liu (IECL, université de Lorraine)

Description

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.