Projections of measures on nilpotent orbits and asymptotic multiplicities of \(K\)-types in rings of regular functions. I (Q1919865)

Let \(G\) be the adjoint group of a real semisimple Lie algebra \({\mathfrak g}\) and \(K\) a maximal compact subgroup of \(G\). Then the complexification \(K_\mathbb{C}\) of \(K\) acts on \({\mathfrak p}^*_\mathbb{C}\), the complexified cotangent space of \(G/K\) at \(eK\). Given a nilpotent \(K_\mathbb{C}\) orbit \({\mathcal O}\) in \({\mathfrak p}^*_\mathbb{C}\), the author studies the asymptotic behavior of \(K\)-type multiplicities in the coordinate ring of the Zariski closure of \({\mathcal O}\). D. Vogan has conjectured that this asymptotic behavior is completely determined by the canonical Liouville measure on the real nilpotent coadjoint orbit \({\mathcal O}_R\) attached to \({\mathcal O}\) by the Sekiguchi correspondence. The author proves this conjecture for Richardson orbits in the complex case and even orbits in general; he also cites some supporting evidence for the general case. The proofs consist largely of explicit calculations with distributions; \textit{J. Sengupta}'s earlier computation of the pushforward of Liouville measure [J. Funct. Anal. 84, 215-225 (1989; Zbl 0729.22020)] and a desingularization of a complex nilpotent orbit also play important roles.