Reduction algebra and differential operators on Lie groups

Abstract

Let $G$ be a connected and simply connected Lie group with Lie algebra $\mathfrak{g }$ of finite dimension. Let $\mathfrak{h }\subset \mathfrak{g }$ be a subalgebra, ${\lambda }$ a character of $\mathfrak{h }$, $\rho$ the trace of the adjoint representation and $\epsilon$ a formal parameter. We prove that the reduction algebra $H^0(\mathfrak{h }^\bot ,\lambda ,\mathfrak{q },\epsilon )$ of Cattaneo–Felder for the Poisson manifold $\mathfrak{g }^*$ and the coisotropic submanifold $-\lambda +\mathfrak{h }^\bot$ is isomorphic with the algebra $U(\mathfrak{g },\mathfrak{h },\lambda +\rho ,\epsilon )^\mathfrak{h }$ of $\mathrm{ad }\mathfrak{h }$-invariant differential operators on $G/H$. At the last section we prove further results relating various deformations and specializations at $\epsilon =1$ of these algebras.