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March 29, 2026Open Access

Representations Associated to Small Nilpotent Orbits for Real Spin Groups

Key Points

The study aims to compare K-structures of unipotent representations and regular sections on nilpotent orbits for real spin groups.
Examined the K-structure of unipotent representations.
Analyzed regular sections of bundles on nilpotent orbits.
Considered nilpotent orbit parameterization and computed K-spectra of certain real forms.
Verified completeness of genuine unipotent representations using coherent continuation representation.
Identified specific unipotent representations linked to nilpotent orbits.
Demonstrated completeness of the identified representations.
Matched K-spectra with K-types of genuine unipotent representations.

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 G 0 = Spin(a, b) with a + b = 2n , the nonlinear double cover of Spin(a, b) , and let K = Spin(a, C) Spin(b, C) be the complexification of the maximal compact subgroup of G 0 .We consider the nilpotent orbit 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 K -spectra of the regular functions on certain real forms O of O c transforming according to appropriate characters under C K (O) , and then match them with the K -types of the genuine unipotent representations.The results provide instances for the orbit philosophy.

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