Small spherical nilpotent orbits and K-types of Harish Chandra modules

2006/12/31 by Donald R. King, King, Donald R.
Mathematics · #14R20 #22E46 #53D20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:14R20 #msc:22E46 #msc:53D20

paper · pdf · doi:10.48550/arxiv.math/0701034

8 pages

arxiv created 2006/12/31 · openalex publication_date 2006/12/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 bar(O is its Zariski closure in p_C. We study the K-type decomposition of the ring of regular functions on bar(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 bar(O).

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