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Lie Groups and Representation Theory
Seminar information archive ～05/22｜Next seminar｜Future seminars 05/23～
| Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|
2011/11/22
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Takayuki Okuda (東京大学大学院 数理科学研究科)
Smallest complex nilpotent orbit with real points (JAPANESE)
Takayuki Okuda (東京大学大学院 数理科学研究科)
Smallest complex nilpotent orbit with real points (JAPANESE)
[ Abstract ]
Let $\\mathfrak{g}$ be a non-compact simple Lie algebra with no complex
structures.
In this talk, we show that there exists a complex nilpotent orbit
$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ in
$\\mathfrak{g}_\\mathbb{C}$ ($:=\\mathfrak{g} \\otimes \\mathbb{C}$)
containing all of real nilpotent orbits in $\\mathfrak{g}$ of minimal
positive dimension.
For many $\\mathfrak{g}$, the orbit
$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is just the
complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.
However, for the cases where $\\mathfrak{g}$ is isomorphic to
$\\mathfrak{su}^*(2k)$, $\\mathfrak{so}(n-1,1)$, $\\mathfrak{sp}(p,q)$,
$\\mathfrak{e}_{6(-26)}$ or $\\mathfrak{f}_{4(-20)}$,
the orbit $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is not
the complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.
We also determine $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$
by describing the weighted Dynkin diagrams of these for such cases.
Let $\\mathfrak{g}$ be a non-compact simple Lie algebra with no complex
structures.
In this talk, we show that there exists a complex nilpotent orbit
$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ in
$\\mathfrak{g}_\\mathbb{C}$ ($:=\\mathfrak{g} \\otimes \\mathbb{C}$)
containing all of real nilpotent orbits in $\\mathfrak{g}$ of minimal
positive dimension.
For many $\\mathfrak{g}$, the orbit
$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is just the
complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.
However, for the cases where $\\mathfrak{g}$ is isomorphic to
$\\mathfrak{su}^*(2k)$, $\\mathfrak{so}(n-1,1)$, $\\mathfrak{sp}(p,q)$,
$\\mathfrak{e}_{6(-26)}$ or $\\mathfrak{f}_{4(-20)}$,
the orbit $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is not
the complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.
We also determine $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$
by describing the weighted Dynkin diagrams of these for such cases.
