Kavli Affiliate: Toshiyuki Kobayashi
| First 5 Authors: Toshiyuki Kobayashi, Birgit Speh, , ,
| Summary:
We present a new approach to symmetry breaking for pairs of real forms of
$(GL(n, mathbb{C}), GL(n-1, mathbb{C}))$. While translation functors are a
useful tool for studying a family of representations of a single reductive
group $G$, when applied to a pair of groups $G supset G’$,translation functors
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 lqlq{fences
for the interlacing pattern}rqrq,which provides a refinement of the usual
notion of lqlq{walls for Weyl chambers}rqrq. We then present a theorem that
states that multiplicity is constant unless these lqlq{fences}rqrq are
crossed. This general theorem is illustrated with examples of both tempered and
non-tempered representations. Additionally,we provide a new non-vanishing
theorem of period integrals for pairs of reductive symmetric spaces,which is
further strengthened through this approach.
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