Abstract:Let $G$ be a connected and simply connected nilpotent Lie group, $K$ an analytic subgroup of $G$ and $\pi$ an irreducible unitary representation of $G$ whose coadjoint orbit of $G$ is denoted by $\Omega (\pi )$. Let $\mathscr U(\mathfrak g)$ be the enveloping algebra of ${\mathfrak g}_{\mathbb C}$, $\mathfrak g$ designating the Lie algebra of $G$. We consider the algebra $D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )\right )^K$ of the $K$-invariant elements of $\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )$. It turns out that this algebra is commutative if and only if the restriction $\pi |_K$ of $\pi$ to $K$ has finite multiplicities (cf. Baklouti and Fujiwara [J. Math. Pures Appl. (9) 83 (2004), pp. 137-161]). In this article we suppose this eventuality and we provide a proof of the polynomial conjecture asserting that $D_{\pi }(G)^K$ is isomorphic to the algebra $\mathbb C[\Omega (\pi )]^K$ of $K$-invariant polynomial functions on $\Omega (\pi )$. The conjecture was partially solved in our previous works (Baklouti, Fujiwara, and Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]; J. Lie Theory 29 (2019), pp. 311-341).

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中文翻译：

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多项式猜想对幂零李群表示限制的证明

摘要：令$G$是一个连通且简单连通的幂零李群，$K$是$G$的解析子群，$\pi$是$G$的不可约酉表示，$G$的共伴轨道记为$ \欧米茄（\pi）$。令$\mathscr U(\mathfrak g)$ 为${\mathfrak g}_{\mathbb C}$ 的包络代数，$\mathfrak g$ 表示$G$ 的李代数。我们考虑 $K$- 的代数 $D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )\right )^K$ $\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )$ 的不变元素。事实证明，这个代数是可交换的当且仅当 $\pi$ 到 $K$ 的限制 $\pi |_K$ 具有有限多重性（参见 Baklouti 和 Fujiwara [J. Math. Pures Appl. (9) 83 （2004 年），第 137-161 页]）。在本文中，我们假设这种可能性，并提供多项式猜想的证明，断言 $D_{\pi }(G)^K$ 与代数 $\mathbb C[\Omega (\pi )]^K$ 同构$\Omega (\pi )$ 上的 $K$-不变多项式函数。该猜想在我们之前的作品中得到了部分解决（Baklouti、Fujiwara 和 Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]；J. Lie Theory 29 (2019), pp. 311-341） .

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