Generic simplicity of quantum Hamiltonian reductions

Akaki Tikaradze

doi:10.5802/crmath.214

Abstract

Let a reductive group G act on a smooth affine complex algebraic variety X. Let 𝔤 be the Lie algebra of G and μ:T * (X)→𝔤 * be the moment map. If the moment map is flat, and for a generic character χ:𝔤→ℂ, the action of G on μ -1 (χ) is free, then we show that for very generic characters χ the corresponding quantum Hamiltonian reduction of the ring of differential operators D(X) is simple.

Highlights

We will assume that this map is flat, and for generic G-invariant character χ ∈ g∗ the action of G on μ−1(χ) is free
In what follows by a very generic subset we mean a complement of a union of countably many proper closed Zariski subsets
We recall that given a ring R such that p is not a zero divisor, the center of its reduction modulo p, Rp = R/pR acquires a natural Poisson bracket, to be referred to as the reduction modulo p Poisson bracket, defined as follows

Summary

Introduction

In what follows by a very generic subset we mean a complement of a union of countably many proper closed Zariski subsets. Let a reductive algebraic group G act on a smooth affine algebraic variety X over C. Let μ : T ∗(X ) → g∗ be the corresponding moment map. We will assume that this map is flat, and for generic G-invariant character χ ∈ g∗ the action of G on μ−1(χ) is free.

Results

Conclusion