Symmetry Breaking under Translations. AIM-IHP seminar. Institut Henri Poincaré, France, 11 February 2025.

I will discuss symmetry breaking for pairs of real forms of $(GL(n, \mathbb{C}), GL(n − 1, \mathbb{C}))$. We introduce the concept of “fences for the interlacing pattern,” which refines the usual notion of “walls for Weyl chambers.” We then present a theorem stating that multiplicity remains constant unless we cross these “fences.” This approach is illustrated with examples of both tempered and non-tempered representations, along with a non-vanishing theorem of period integrals for pairs of reductive symmetric spaces.

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© Toshiyuki Kobayashi