Lie群論・表現論セミナー

過去の記録 ~05/02次回の予定今後の予定 05/03~

開催情報 火曜日 16:30~18:00 数理科学研究科棟(駒場) 126号室
担当者 小林俊行
セミナーURL https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2024年11月27日(水)

13:30-14:30   数理科学研究科棟(駒場) 122号室
日仏数学拠点 FJ-LMI セミナーと合同
藤原英徳 氏 (OCAMI, 近畿大学)
Inductions and restrictions of unitary representations for exponential solvable Lie groups (English)
[ 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 $\tau=\operatorname{ind}_H^G \chi$ 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 $\chi_f(\exp X)=e^{if(X)}$ $(X∈{\mathfrak{h}})$ with a certain $f∈{\mathfrak{g}}^{\ast}$ verifying $f([{\mathfrak{h}}, {\mathfrak{h}}])=\{0\}$. Let $S({\mathfrak{g}})$ be the symmetric algebra of ${\mathfrak{g}}$ and ${\mathfrak{a}}_{\tau}=\{X+\sqrt{-1} f(X) ; X∈{\mathfrak{h}}\}$. We regard $S({\mathfrak{g}})$ as the algebra of polynomial functions on ${\mathfrak{g}}^{\ast}$ by $X(\ell)=\sqrt{-1} \ell(X)$ for $X∈{\mathfrak{g}}$, $\ell ∈{\mathfrak{g}}^{\ast}$. 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 $\Gamma_{\tau}=\{ℓ \in {\mathfrak{g}}^{\ast}:\ell(X)=f(X),X \in {\mathfrak{h}}\}$ of ${\mathfrak{g}}^{\ast}$. 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.