Tuong Ton-That: A Generalized Poincaré Theorem for Dual Lie Transformation Groups


Submission: 2001, Jun 13

Let $k$ and $n$ be integers such that $k>2n>0$. Let $M$ be the complex analytic manifold defined by $M=\{x\in\mathbb{C}^{n\times k}:xx^{t}% =0,\;\operatorname*{rank}$ $(x)=n\}$. Let $G=\mathrm{SO}(k,\mathbb{C})$ and $G^{\prime}=\mathrm{GL}(n,\mathbb{C})$, then Witt's theorem on quadratic forms implies that $G$ is a maximal connected Lie group acting transitively on $M$ by right multiplication. Also, $G^{\prime}$ is a maximal connected Lie group acting freely on $M$ by left multiplication. If $f\in C^{\infty}(M)$, $x\in M$, $g\in G$, and $g^{\prime}\in G^{\prime}$ define $R(g)f$ (resp.~$L(g^{\prime})f$) by \[ (R(g)f)(x)=f(xg)\text{\quad and\quad}(L(g)f)(x)=f(g^{-1}x). \] If $\mathcal{D}^{\omega}(M)$ denotes the algebra of all analytic differential operators on $M$ then an element $D\in\mathcal{D}^{\omega}(M)$ is called right (resp.~left)-invariant if $DR(g)=R(g)D$, $\forall\,g\in G$ (resp.~$DL(g^{\prime})=L(g^{\prime})D,$ $\forall\,g^{\prime}\in G^{\prime}$).

THEOREM: Let $\mathcal{D}_{l}^{\omega}(M)$ (resp.~$\mathcal{D}_{r}^{\omega }(M)$) denote the subalgebra of $\mathcal{D}^{\omega}(M)$ of all left (resp.~right)-invariant analytic differential operators on $M$. Let $\widetilde{\mathcal{U}}(\frak{g})$ (resp.$~\widetilde{\mathcal{U}}% (\frak{g}^{\prime})$) denote the universal enveloping algebra generated by the infinitesimal action of $R(g)$ (resp.~$L(g^{\prime})$). Then we have \[ \mathcal{D}_{l}^{\omega}(M)=\widetilde{\mathcal{U}}(\frak{g})\text{ and }\mathcal{D}_{r}^{\omega}(M)=\widetilde{\mathcal{U}}(\frak{g}^{\prime })\text{.}% \] Moreover, the commutant of $\mathcal{D}_{l}^{\omega}(M)$ in $\mathcal{D}% ^{\omega}(M)$ is $\mathcal{D}_{r}^{\omega}(M)$, and vice-versa.

This theorem also holds for other types of dual Lie transformation groups acting on analytic manifolds.

2000 Mathematics Subject Classification: Primary 15A63, 16S32; Secondary 16S30, 14L35.

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