Quantization of algebraic cones and Vogan’s conjecture

Abstract

Let C be a complex algebraic cone, provided with an action of a compact Lie group K. The symplectic form of the ambient complex Hermitian space induces on the regular part of C a symplectic form. Let k be the Lie algebra of K. Let f : C → k∗ be the Mumford moment map, that is f(v)(X) = i(v,Xv), for X ∈ k and v ∈ C. The space R(C) of regular functions on C is a semi-simple representation of K. In this article, with the help of the moment map, we give some quantitative informations on the decomposition of R(C) in irreducible representations of K. For λ a dominant weight, let m(λ) be the multiplicity of the representation of highest weight λ in R(C). Then, if the moment map f : C → k∗ is proper, multiplicities m(λ) are finite and with polynomial growth in λ. Furthermore, the study of the pushforward by f of the Liouville measure gives us an asymptotic information on the function m(λ) . For example, in the case of a faithful torus action, the pushforward of the Liouville measure by the moment map is a locally polynomial homogeneous function `(λ) on the polyhedral cone f(C) ⊂ t∗, while the multiplicity function m(λ) for large values of λ is given by the restriction to the lattice of weights of a quasipolynomial function, with highest degree term equal to `(λ). If O is a nilpotent orbit of the coadjoint representation of a complex Lie group G, we show that the pushforward on k∗ of the G-invariant measure on O is the same that the pushforward of the Liouville measure on O associated to the symplectic form of the ambient complex vector space. Thus, this establishes for the case of complex reductive groups the relation, conjectured by D. Vogan, between the Fourier transform of the orbit O and multiplicities of the ring of regular functions on O.