Pseudo-dual pairs and branching of Discrete Series

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 H0, 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.