## Geometry Seminar - Abstracts

### Talk

Thursday 24 May 2018, 16:00-17:00 in HG03.085

**Maarten van Pruijssen** (Paderborn)

*Inverting the branching rules in a multiplicity free setting*

### Abstract

Branching rules for compact groups describe how an irreducible representation
decomposes upon restricting to a compact subgroup.
In general, the branching rule can be described by an alternating sum over a
Weyl group of counting functions depending on roots and weights.
In certain multiplicity free examples, one can rewrite these complicated
formulas into "interlacings conditions" for the weights of the irreducible
representations.
Assuming a multiplicity freeness condition, we will discuss the inversion of
the branching rules: fix an irreducible representation of the subgroup and
determine in which irreducible representations of the overgroup it occurs upon
restriction.
The multiplicity free setting at hand allows us to invoke the theory of
spherical varieties to solve this problem. We recover the Cartan-Helgason
theorem for symmetric spaces, some of the classical branching rules and new
inverted branching rules that are relevant for harmonic analysis of vector
bundles over compact Riemann symmetric pairs.
This is joint work with Guido Pezzini (La Sapienza, Rome).

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