Method of quantum characters in equivariant quantization

Abstract

Let $G$ be a reductive Lie group, $\g$ its Lie algebra, and $M$ a $G$-manifold. Suppose $\A_h(M)$ is a $\U_h(\g)$-equivariant quantization of the function algebra $\A(M)$ on $M$. We develop a method of building $\U_h(\g)$-equivariant quantization on $G$-orbits in $M$ as quotients of $\A_h(M)$. We are concerned with those quantizations that may be simultaneously represented as subalgebras in $\U^*_h(\g)$ and quotients of $\A_h(M)$. It turns out that they are in one-to-one correspondence with characters of the algebra $\A_h(M)$. We specialize our approach to the situation $\g=gl(n,\C)$, $M=\End(\C^n)$, and $\A_h(M)$ the so-called reflection equation algebra associated with the representation of $\U_h(\g)$ on $\C^n$. For this particular case, we present in an explicit form all possible quantizations of this type; they cover symmetric and bisymmetric orbits. We build a two-parameter deformation family and obtain, as a limit case, the $\U(\g)$-equivariant quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits.

Replaced by the journal version, 28 pages, AMS LaTeX

References in corpus (1)

• $\U_q(sl(n))$-invariant quantization of symmetric coadjoint orbits via reflection equation algebra

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