Strict quantization of coadjoint orbits

Abstract

A strict quantization of a compact symplectic manifold $S$ on a subset $I\subseteq\R$, containing 0 as an accumulation point, is defined as a continuous field of $C^*$-algebras $\{A_{\hbar}\}_{\hbar\in I}$, with $A_0=C_0(S)$, and a set of continuous cross-sections $\{Q(f)\}_{f\in C^{\infty}(S)}$ for which $Q_0(f)=f$. Here $Q_{\hbar}(f^*)=Q_{\hbar}(f)^*$ for all $\hbar\in I$, whereas for $\hbar\to 0$ one requires that $i[Q_{\hbar}(f),Q_{\hbar}(g)]/\hbar\to Q_{\hbar}(\{f,g\})$ in norm. We discuss general conditions which guarantee that a (deformation) quantization in a more physical sense leads to one in the above sense. Using ideas of Berezin, Lieb, Simon, and others, we construct a strict quantization of an arbitrary integral coadjoint orbit $O_{\lm}$ of a compact connected Lie group $G$, associated to a highest weight $\lm$. Here $I=0\cup 1/\N$, so that $\hbar=1/k$, $k\in\N$, and $A_{1/k}$ is defined as the $C^*$-algebra of all matrices on the finite-dimensional Hilbert space $V_{k\lm}$ carrying the irreducible representation $U_{k\lm}(G)$ with highest weight $k\lm$. The quantization maps $Q_{1/k}(f)$ are constructed from coherent states in $V_{k\lm}$, and have the special feature of being positive maps.