By Ali Baklouti
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 $\mathcal U(\mathfrak g)$ be the enveloping algebra of $\mathfrak g_{\mathbb C}$, $\mathfrak g$ designating the Lie algebra of $G$. We consider the algebra $D_{\pi}(G)^K\simeq (\mathcal U(\mathfrak g)/ \ker \pi)^K$ of the $K$-invariant elements of $\mathcal U(\mathfrak g)/ \ker \pi$. It turns out that this algebra is commutative if and only if the restriction $\pi|K$ of $\pi$ to $K$ has finite multiplicities (A. Baklouti–H. Fujiwara 2004). In this lecture, I will talk about 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 $\Omega(\pi)$, provided that $\pi|K$ is of finite multiplicities. This is a joint work with H. Fujiwara and J. Ludwig.
By Salem Ben Said
By Patrick Delorme
By Salma Nasrin
By Genkai Zhang
By Andreas Juhl
By Reiko Miyaoka
By Ali Baklouti
By Yoshiki Oshima
By Dan Barbasch
By Kazuki Kannaka
By Jan Frahm
By Angela Pasquale
By Jean-Louis Clerc
By Ronnie Pavlov
By Tamara Kucherenko
