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

In this talk, I will discuss three Cartan subalgebras (or root systems) related to the branching problem of reductive Lie groups. One Cartan subalgebra describes complexity of an embedding of $G$-varieties. This is related to intertwining operators (symmetry breaking operators). The others are defined by the annihilators of $\mathfrak{g}$-modules or their non-zero vectors. They are related to the shape of the (continuous) spectrum. I will also discuss a relation between the Cartan subalgebras and the moment maps for Hamiltonian actions.