Symmetry Breaking under Translation. Representation Theory XIX. Dubrovnik, Croatia, 22-28 June 2025. Organizers: Dražen Adamović, Tomoyuki Arakawa, Andrej Dujella, Matija Kazalicki, Hrvoje Kraljević, Antun Milas, Filip Najman, Pavle Pandžić, Ana Prlić, Mirko Primc and David Vogan.

We consider the restriction of irreducible representations of a real reductive group G to a subgroup H. In general, the multiplicity in the branching law can be infinite, even when H is a maximal reductive subgroup, such as in the case where $(G,H)=(GL(p+q),GL(p) ×GL(q))$. In this talk, I plan to start with general results on multiplicities and present criteria for uniform boundedness. Then, we introduce a concept of “fences” describing interleaving patterns, which refine the usual notion of “walls” for Weyl chambers. A theorem will be presented, stating that the multiplicity remains constant unless these “fences” are crossed. If time permits, I will mention some applications to new branching laws.
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