# Pseudo-dual pairs and branching of Discrete Series

## TL;DR

The paper investigates Discrete Series representations of semisimple Lie groups with admissible branching to symmetric subgroups, introducing pseudo-dual pairs and explicit branching laws, with applications to computational algorithms like Atlas.

## Contribution

It introduces the concept of pseudo-dual pairs for symmetric subgroups and develops explicit branching laws, connecting to symmetry breaking and computational methods.

## Key findings

- Explicit examples of branching laws provided.
- Links established to symmetry breaking and holographic operators.
- Method suitable for computer algorithms like Atlas.

## Abstract

For a semisimple Lie group $G$, we study Discrete Series representations with admissible branching to a symmetric subgroup $H$. This is done using a canonical associated symmetric subgroup $H_0$, forming a pseudo-dual pair with $H$, and a corresponding branching law for this group with respect to its maximal compact subgroup. This is in analogy with either Blattner's or Kostant-Heckmann multiplicity formulas, and has some resemblance to Frobenius reciprocity. We give several explicit examples and links to Kobayashi-Pevzner theory of symmetry breaking and holographic operators. Our method is well adapted to computer algorithms, such as for example the Atlas program.