By Birgit Speh
Appears in collection : Representation theory and analysis on homogeneous spaces / Théorie des représentations et Analyse sur les espaces homogènes
I am presenting joint work with Professor Kobayashi. We worked out a new approach to symmetry breaking for pairs of real forms of $(G L(n, \mathbb{C}), G L(n-1, \mathbb{C}))$.
Translation functors are powerful tools for studying families of representations of a single reductive group $G$. However, when applied to a pair of groups $G \supset G^{\prime}$, they can significantly alter the nature of symmetry breaking between the representations of $G$ and $G^{\prime}$, even within the same Weyl chamber of the direct product group $G \times G^{\prime}$.
We introduce the concept of fences for the interleaving pattern, which provides a refinement of the usual notion of "walls of Weyl chambers. We then establish a theorem stating that the multiplicity remains constant unless these fences are crossed, together with a new general vanishing theorem for symmetry breaking.
By Toshiyuki Kobayashi
By Sundaram Thangavelu
