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 ℓ(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 ℓ(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.
