Kavli Affiliate: Toshiyuki Kobayashi
| Summary:
We present a new approach to symmetry breaking for pairs of real forms of $(GL(n, mathbbC), GL(n-1, mathbbC))$. Translation functors are powerful tools for studying families of representations of a single reductive group $G$. However, when applied to a pair of groups $G supset G’$, they can significantly alter the nature of symmetry breaking between the representations of $G$ and $G’$, even within the same Weyl chamber of the direct product group $G times G’$.
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.
These general results are illustrated with examples involving both tempered and non-tempered representations. In addition, we present a new non-vanishing theorem for period integrals for pairs of reductive symmetric spaces, which is further strengthened by this approach.
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