On a conjecture of Guillemin and Sternberg in geometric quantization of multiplicity-free symplectic spaces (Q1176455)

Let a compact group \(K\) act transitively on a symplectic manifold \(X\). The representation of \(K\) in the quantization \(X_ q\) of \(X\) is called multiplicity-free if the multiplicity of every irreducible representation of \(K\) in \(X_ q\) is one or zero. Guillemin and Sternberg proved that this property is equivalent to the statement that the Poisson algebra of \(K\)-invariant functions on \(X\) is abelian, and conjectured that the equivalence remains true for Poisson actions. The author proves a modified version of the conjecture in the compact Kähler case, in terms of holomorphic sections of the powers of a line bundle.