Small Spherical Nilpotent Orbits and K-types of Harish Chandra Modules

Let G be a connected linear semisimple Lie group with Lie algebra g and maximal compact subgroup K. Let K C → Aut(p C ) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that O is a nilpotent K C -orbit in p C , and O is its Zariski closure in p C . We study the K-type decomposition of the ring of regular functions on O when O is spherical and “small”. We also show that this decomposition gives the asymptotic directions of K-types in any irreducible (g C , K)-module whose associated variety is O. 1. A desingularisation of the closure of a nilpotent K C -orbit in p C Let G be a connected linear semisimple Lie group with Lie algebra g and maximal compact subgroup K. Let K C → Aut(p C ) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that O is a nilpotent K C -orbit in p C , and O is its Zariski closure in p C . O n denotes the normalization of O. If W is an algebraic variety, R[W ] denotes the ring of regular functions on W . We recall some facts related to vector bundles over a homogeneous space M / P for a complex reductive group M and parabolic subgroup P . Let V be a finite dimensional P -module. Then, M×QV = M × V / ∼, where the equivalence relation “∼” is defined by: (gy, v) ∼ (g, y ·v), for g ∈ M , y ∈ P and v ∈ V . The equivalence relation “∼” can be seen as arising from a right action of P on M × V defined as follows: (g, v) · y = (gy, y · v). Then it is easy to check that (g, v) · (y1y2) = [(g, v) · y1] · y2. Also, note that (g, v) ∼ (g, v) · y, for all g ∈ M, v ∈ V, y ∈ P . The equivalence class of (g, v) under ∼ is denoted [g, v]. A key fact is that: R[M ×P V ] = R[M × V ] P , where the P action on R[M × V ] is defined from the P action on M × V : if k ∈ R[M × V ], and y ∈ P , then (k · y)(g, v) = k((g, v) · y) = k(gy, y · v). The equality of the rings above is implemented as follows. If f ∈ R[M ×P V ], then define f̃ on M × V , by f̃(g, v) = f([g, v]). Since f([g, v]) = f([gyy, v]) = f([gy, y · v]), f̃ ∈ R[M × V ] . On the other hand if h ∈ R[M × V ] , then ĥ([g, v]) := h(g, v) is well defined and ĥ ∈ R[M ×P V ]. 2000 Mathematics Subject Classificationn. Primary 22E46; Secondary 14R20, 53D20.