Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

Abstract

We construct a unique G-equivariant graded star product on the algebra $S(g)/I$ of polynomial functions on the minimal nilpotent coadjoint orbit $\Omin$ of G where G is a complex simple Lie group and $g\neq\sl_2(C)$. This strengthens the result of Arnal, Benamor, and Cahen. Our main result is to compute, for G classical, the star product of a momentum function $\mu_x$ with any function f. We find $\mu_x\star f=\mu_xf+\half\{\mu_x,f\}t+\Lambda^x(f)t^2$. For $\g$ different from $sp_n(\C)$, $\Lambda^x$ is not a differential operator. Instead $\Lamda^x$ is the left quotient of an explicit order 4 algebraic differential operator $D^x$ by an order 2 invertible diagonalizable operator. Precisely, $\Lambda^x=-{1/4}\frac{1}{E'(E'+1)}D^x$ where $E'$ is a positive shift of the Euler vector field. Thus $\mu_x\star f$ is not local in f. Using $\star$ we construct a positive definite hermitian inner product on $Sg/I$. The Hilbert space completion of $Sg/I$ is then a unitary representation of $G$. This quantizes $\Omin$ in the sense of geometric quantization and the orbit method.