日仏数学拠点FJ-LMIセミナー

担当者	小林俊行, ミカエル　ペブズナー

2024年11月27日(水)

13:30-14:30 数理科学研究科棟(駒場) 122号室
藤原英徳氏 (OCAMI, 近畿大学)
Inductions and restrictions of unitary representations for exponential solvable Lie groups. (英語)

[ 講演概要 ]
Let

$G = \exp \mathfrak g$ be a connected and simply connected real nilpotent Lie group with Lie algebra

$\mathfrak g$ ,

$H = \exp \mathfrak h$ an analytic subgroup of

$G$ with Lie algebra

$\mathfrak h$ ,

$\chi$ a unitary character of

$H$ and

$\tau = \text{ind}_H^G \chi$ the monomial representation of

$G$ induced from

$\chi$ . Let

$D_{\tau}(G/H)$ be the algebra of the

$G$ -invariant differential operators on the line bundle over

$G/H$ associated to the data

$(H,\chi)$ . We denote by

$C_{\tau}$ the center of

$D_{\tau}(G/H)$ . We know that

$\chi$ is written as

${\chi}_f$ , where

$\chi_f(\exp X) = e^{if(X)} (X \in \mathfrak h)$ with a certain

$f \in {\mathfrak g}^*$ verifying

$f([\mathfrak h,\mathfrak h]) = \{0\}$ . Let

$S(\mathfrak g)$ be the symmetric algebra of

$\mathfrak g$ and

${\mathfrak a}_{\tau} = \{X + \sqrt{-1}f(X) ; X \in \mathfrak h\}.$ We regard

$S(\mathfrak g)$ as the algebra of polynomial functions on

${\mathfrak g}^*$ by

$X(\ell) = \sqrt{-1}\ell(X)$ for

$X \in \mathfrak g, \ell \in {\mathfrak g}^*$ . Now,

$S(\mathfrak g)$ possesses the Poisson structure

$\{,\}$ well determined by the equality

$\{X,Y\} = [X,Y]$ if

$X, Y$ are in

$\mathfrak g$ . Let us consider the algebra

$(S(\mathfrak g)/S(\mathfrak g)\overline{{\mathfrak a}_{\tau}})^H$ of the

$H$ -invariant polynomial functions on the affine subspace

${\Gamma}_{\tau} = \{\ell \in {\mathfrak g}^* : \ell(X) = f(X), X \in \mathfrak h\}$ of

${\mathfrak g}^*$ . This inherits the Poisson structure from

$S(\mathfrak g)$ . We denote by

$Z_{\tau}$ its Poisson center. Michel Duflo asked: the two algebras

$C_{\tau}$ and

$Z_{\tau}$ , are they isomorphic? Here we provide a positive answer to this question.

[ 講演参考URL ]
https://fj-lmi.cnrs.fr/seminars/

このページのトップへ

東京大学大学院数理科学研究科理学部数学科・理学部数学科

〒153-8914　東京都目黒区駒場3-8-1　TEL:03-5465-7001（代表受付）　FAX：03-5465-7011（代表受付）