[geometry-ml:05342] セミナーのご案内

[Tohru Morimoto]

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岡数学研究所
森本　徹


Seminar in Oka Mathematical Institute 

Date:       September 25 Monday, 2023
Place:      Oka Mathematical Institute, Nara Women's University

Speakers and Titles

10:30-11:30 Boris Doubrov( Belarus State University)
 Equivalence of bracket-generating vector distributions via control theory
              
 13:30-14:30 Toshihisa Kubo( Ryukoku University)
 On the equivariant differential operators for SL(3,R) with maximal parabolic subgroup

 15:00-16:00  Masanori Adachi(Shizuoka University)
  TBA
              
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Titles and Abstracts

 B.Doubrov
Title:    Equivalence of bracket-generating vector distributions via control theory
Abstract: We canonically associate a pseudo-product structure with any bracket-generating distribution. Under some additional non-degeneracy conditions one can use the linearization principle and study these structures via systems of linear ODEs whose solution space possesses an invariant symplectic form.
 
The symbol of such linear systems is described by a homogeneous element of degree -1 in the Lie algebra sp(2n,R) equipped with an arbitrary parabolic grading. We show how such elements can be effectively classified.
 
In case of (2,3,5) distributions this construction leads to a pseudo-product structure modeled by the parabolic homogeneous space G_2/B  where B is the Borel subgroup.


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T.Kubo

 Title:  On the equivariant differential operators for SL(3,R) with maximal parabolic subgroup
 Abstract: Let $G \supset P$ be a real simple Lie group and a parabolic subgroup, respectively. For finite dimensional representations $V_i$ ($i=1,2$) of $P$, let $\mathcal{V}_i \to G/P$ denote the $G$-equivariant homogeneous vector bundle $\mathcal{V}_i$ over $G/P$ with fiber $V_i$. Differential operators $\mathcal{D}\colon C^\infty(G/P, \mathcal{V}_1) \to  C^\infty(G/P, \mathcal{V}_2)$ are said to be \emph{equivariant} (or \emph{covariant} or \emph{intertwining}) if $\mathcal{D}$ is equivariant under the actions of $G$ on $C^\infty(G/P, \mathcal{V}_i)$ $(i=1,2)$. A classical example of such an operator is the wave operator $\square_{3,1}$ on the Minkowski space $\mathbb{R}^{3,1}$ of signature $(3,1)$. Equivariant differential operators are important objects in both representation theory and parabolic geometry.
 
In this talk we shall discuss a classification and explicit construction of equivariant differential operators for the following setting:
\begin{equation*}
(G,P; V_1, V_2) = (SL(3,\mathbb{R}), P_{1,2}; F_n, F_m).
\end{equation*}
Here, $P_{1,2}$ is the maximal parabolic subgroup corresponding to the partition $(1,2)$, and $F_n$ (resp.\ $F_m$) is the finite dimensional irreducible representation of $P_{1,2}$ of dimension $n$ (resp.\ $m$).  This is based on joint work in progress with Bent Orsted.
 
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