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Lie Groups and Representation Theory
Seminar information archive ～06/30｜Next seminar｜Future seminars 07/01～
| Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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2025/07/08
15:30-16:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Koichi Arashi (Tokyo Gakugei University)
On integral representations of reproducing kernels on quasi-symmetric Siegel domains
Koichi Arashi (Tokyo Gakugei University)
On integral representations of reproducing kernels on quasi-symmetric Siegel domains
[ Abstract ]
L.\ Schwartz established the foundational theory of reproducing kernels in the 1960s.
Around the same time, S.\ G.\ Gindikin obtained an explicit integral representation of the Bergman kernel for the Siegel domain of the second kind $\mathcal{S}(\Omega,Q)\subset U_{\mathbb C}\times V$.
This formula suggests that the set of irreducible unitary representations of the generalized Heisenberg group $G^{V}=U\rtimes V$ realized on this domain is embedded in the unitary dual of the group.
Such a notion of multiplicity-freeness property has since been reconsidered from a complex-geometric standpoint, motivated by Huckleberry–Wurzbacher's study of ``coisotropic actions'' and by T.\ Kobayashi's introduction of ``visible actions'', and its understanding continues to deepen.
In this talk, we focus on a quasi-symmetric Siegel domain, and for a real subspace $W\subset V$, study the representations of the subgroup $G^{W}=U\rtimes W$.
We show that the multiplicity-freeness property can be characterized both by geometric features of the group action and by the multiplicity-free irreducible decomposition of the unitary representation on the Bergman space.
L.\ Schwartz established the foundational theory of reproducing kernels in the 1960s.
Around the same time, S.\ G.\ Gindikin obtained an explicit integral representation of the Bergman kernel for the Siegel domain of the second kind $\mathcal{S}(\Omega,Q)\subset U_{\mathbb C}\times V$.
This formula suggests that the set of irreducible unitary representations of the generalized Heisenberg group $G^{V}=U\rtimes V$ realized on this domain is embedded in the unitary dual of the group.
Such a notion of multiplicity-freeness property has since been reconsidered from a complex-geometric standpoint, motivated by Huckleberry–Wurzbacher's study of ``coisotropic actions'' and by T.\ Kobayashi's introduction of ``visible actions'', and its understanding continues to deepen.
In this talk, we focus on a quasi-symmetric Siegel domain, and for a real subspace $W\subset V$, study the representations of the subgroup $G^{W}=U\rtimes W$.
We show that the multiplicity-freeness property can be characterized both by geometric features of the group action and by the multiplicity-free irreducible decomposition of the unitary representation on the Bergman space.
