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Seminar on Geometric Complex Analysis
Seminar information archive ～10/17｜Next seminar｜Future seminars 10/18～
| Date, time & place | Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Kengo Hirachi, Shigeharu Takayama |
2025/10/20
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Hideyuki Ishi (Osaka Metropolitan Univ.)
A CR-Laplacian type operator for the Silov boundary of a homogeneous Siegel domain (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
Hideyuki Ishi (Osaka Metropolitan Univ.)
A CR-Laplacian type operator for the Silov boundary of a homogeneous Siegel domain (Japanese)
[ Abstract ]
Let $\Sigma$ be the Silov boundary of a homogeneous Siegel domain $D$ on which a Lie group $G$ acts transitively as affine transformations. The CR-structure on $\Sigma$ naturally induced from the ambient complex vector space is non-trivial if and only if $D$ is of non-tube type. In this case, $\Sigma$ is naturally identified with a two-step nilpotent Lie subgroup $N$ of $G$, called a generalized Heisenberg Lie group. Since the CR-structure is invariant under the action of $G$, the CR-cohomology space over $\Sigma$ can be regarded as a $G$-module. We consider unitarization of this presentation of $G$. The kernel of the CR-Laplacian does not give the solution because the natural Riemannian metric on $\Sigma$ is not $G$-invariant, so that the $G$-action does not preserve the space of CR-harmonic forms. Nevertheless, Nomura defined a unitary $G$-action on the space indirectly when $G$ is split solvable. In this talk, we introduce a space of CR-cochains with $G$-invariant inner product defined via the Fourier transform. Then the associated CR-operator is no longer a differential operator, while the kernel of the operator gives a unitarization of the representation of $G$ over the cohomology space.
[ Reference URL ]Let $\Sigma$ be the Silov boundary of a homogeneous Siegel domain $D$ on which a Lie group $G$ acts transitively as affine transformations. The CR-structure on $\Sigma$ naturally induced from the ambient complex vector space is non-trivial if and only if $D$ is of non-tube type. In this case, $\Sigma$ is naturally identified with a two-step nilpotent Lie subgroup $N$ of $G$, called a generalized Heisenberg Lie group. Since the CR-structure is invariant under the action of $G$, the CR-cohomology space over $\Sigma$ can be regarded as a $G$-module. We consider unitarization of this presentation of $G$. The kernel of the CR-Laplacian does not give the solution because the natural Riemannian metric on $\Sigma$ is not $G$-invariant, so that the $G$-action does not preserve the space of CR-harmonic forms. Nevertheless, Nomura defined a unitary $G$-action on the space indirectly when $G$ is split solvable. In this talk, we introduce a space of CR-cochains with $G$-invariant inner product defined via the Fourier transform. Then the associated CR-operator is no longer a differential operator, while the kernel of the operator gives a unitarization of the representation of $G$ over the cohomology space.
https://forms.gle/gTP8qNZwPyQyxjTj8
