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

Abstract

Let G G be a connected and simply connected nilpotent Lie group, K K an analytic subgroup of G G and π \pi an irreducible unitary representation of G G whose coadjoint orbit of G G is denoted by Ω ( π ) \Omega (\pi ) . Let U ( g ) \mathscr U(\mathfrak g) be the enveloping algebra of g C {\mathfrak g}_{\mathbb C} , g \mathfrak g designating the Lie algebra of G G . We consider the algebra D π ( G ) K ≃ ( U ( g ) / ker ⁡ ( π ) ) K D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )\right )^K of the K K -invariant elements of U ( g ) / ker ⁡ ( π ) \mathscr U(\mathfrak g)/\operatorname {ker}(\pi ) . It turns out that this algebra is commutative if and only if the restriction π | K \pi |_K of π \pi to K 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 π ( G ) K D_{\pi }(G)^K is isomorphic to the algebra C [ Ω ( π ) ] K \mathbb C[\Omega (\pi )]^K of K K -invariant polynomial functions on Ω ( π ) \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).