A proof of the polynomial conjecture for restrictions of nilpotent lie groups representations

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).