Quantization of algebraic cones and Vogan's conjecture (Q1392485)

Let \(K\) be a compact group acting by unitary mappings on the complex vector space \(V\) and \(C \subseteq V\) an algebraic \(K\)-invariant cone. Then the regular part of \(C\) inherits a Kähler structure from \(V\) such that the action of \(K\) becomes Hamiltonian with moment map \(f : C \to {\mathbf k}^*\) given by \(f(v)(X)= i (v, X.v)\), \(X \in {\mathbf k}\), \(v \in C\). The main objective of this article are the relations between properties of the moment map \(f\) and the decomposition of the representation of \(K\) on the space \(R(C)\) of regular functions on \(C\). For a dominant weight \(\lambda\) let \(m(\lambda)\) denote the multiplicity of the corresponding irreducible representation of \(K\) in \(R(C)\). It is shown that if the moment map is proper, then the multiplicities \(m(\lambda)\) are finite and of polynomial growth in \(\lambda\). Moreover, the pushforward of the Liouville measure by the moment map \(f\) is a Radon measure on \({\mathbf k}^*\) which provides information on the asymptotic behaviour of the multiplicities. This setup applies in particular to the situation where \(C\) is the closure of a nilpotent orbit \(O\) in the adjoint representation of the complex Lie algebra \({\mathbf g}\) and \(K\) is a connected group whose Lie algebra \({\mathbf k}\) is a real form of \({\mathbf g}\). In this case it is shown that the pushforward of the \(G\)-invariant measure on \(O\) to \({\mathbf k}^*\) by the moment map \(f\) coincides with that of the Liouville measure associated to the symplectic form on \(O \subseteq C\) corresponding to the \(K\)-invariant Kähler structure inherited from the \(K\)-invariant positive definite hermitian form on \({\mathbf g}\). For the case of complex reductive groups, this correspondence has been conjectured by D. Vogan in terms of the Fourier transform of the orbit \(O\) and the asymptotics of the multiplicities. In this sense the general setting described above serves as an intermediate stage relating multiplicities and pushforwards of Liouville measures.