Non-Locality of Equivariant Star Products on $T\ast (R{P}^n)$

Abstract

Lecomte and Ovsienko constructed $S{L}_{n+1}(R)$-equivariant quantization maps $Q_{\lambda }$ for symbols of differential operators on $\lambda$-densities on $\RP^n$. We derive some formulas for the associated graded equivariant star products $sta{r}_{\lambda }$ on the symbol algebra $Pol(T*\RP^n)$. These give some measure of the failure of locality. Our main result expresses (for $n$ odd) the coefficients $C_p$ of $sta{r}_{\lambda }$ when $λ=\half$ in terms of some new $S{L}_{n+1}(C)$-invariant algebraic bidifferential operators $Z_p$ on $T*\CP^n$ and the operators $(E+\frac{n}{2}\pm s{)}^{-1}$ where $E$ is the fiberwise Euler vector field and $s\in \{1,2,...,[\frac{p}{2}]\}$.

6 pages