Quasi-Poisson Modules and Harish-Chandra AD-Modules

Abstract

We introduce the notion of quasi-Poisson modules over Lie-Rinehart pairs and prove that for the Lie-Rinehart pair $(\dot A,\dot\fk)$ in which $\dot A=\bbbc[t_1^{\pm1},\ldots,t_m^{\pm1}]\ot\Lam_n$ and $\dot\fk={\rm Der}(\dot A)$, there is a one-to-one correspondence between simple cuspidal quasi-Poisson modules over $(\dot A,\dot\fk)$ and simple cuspidal Harish-Chndra $A\fk$-modules for $A:=\bbbc[t_0^{\pm1}]\ot \dot A$ and $\fk:={\rm Der}(A).$ We also classify simple cuspidal quasi-Poisson modules over the Lie-Rinehart pair $(\dot A,\dot\fk)$ and show that each such module is a tensor module $\dot A\ot Ω$ for an admissible $\frak{gl}(m+1,n)$-module $Ω$ via a prescribed action.