Representations associated to small nilpotent orbits for real Spin groups

2017/02/16 by Dan Barbasch, Wan-Yu Tsai, Barbasch, Dan +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1702.04841

openalex publication_date 2017/02/16 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

The results in this paper provide a comparison between the K-structure of unipotent representations and regular sections of bundles on nilpotent orbits. Precisely, let \widetildeG₀ =\widetildeSpin(a,b) with a+b=2n, the nonlinear double cover of Spin(a,b), and let \widetildeK=Spin(a, \mathbb C)× Spin(b, \mathbb C) be the complexification of the maximal compact subgroup of \widetildeG₀. We consider the nilpotent orbit \mathcal O_c parametrized by [3 2^2k 1^2n-4k-3] with k>0. We provide a list of unipotent representations that are genuine, and prove that the list is complete using the coherent continuation representation. Separately we compute \widetildeK-spectra of the regular functions on certain real forms \mathcal O of \mathcal O_c transforming according to appropriate characters ψ under C_\widetildeK(\mathcal O), and then match them with the \widetildeK-types of the genuine unipotent representations. The results provide instances for the orbit philosophy.

Related

• Representations associated to small nilpotent orbits for complex Spin groups
• Unipotent representations and the dual pair correspondence
• Unipotent Representations Attached to the Principal Nilpotent Orbit
• Genuine special unipotent representations of spin groups
• Regular functions on covers of nilpotent coadjoint orbits