Representations associated to small nilpotent orbits for complex Spin groups

TL;DR

This paper analyzes the $K$-structure of unipotent representations and regular functions on small nilpotent orbits for complex Spin groups, providing explicit computations and matching these structures.

Contribution

It computes $K$-spectra of functions on small nilpotent orbits and matches them with $K$-types of unipotent representations for complex Spin groups.

Findings

Computed $K$-spectra for small nilpotent orbits.

Matched $K$-spectra with unipotent representations.

Provided explicit descriptions of representations related to nilpotent orbits.

Abstract

This paper provides a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type $D$. Precisely, let $ G_ 0 =Spin(2n,\mathbb C)$ be the Spin complex group viewed as a real group, and $K\cong G_0$ be the complexification of the maximal compact subgroup of $G_0$. We compute $K$-spectra of the regular functions on some small nilpotent orbits $\mathcal O$ transforming according to characters $\psi$ of $C_{ K}(\mathcal O)$ trivial on the connected component of the identity $C_{ K}(\mathcal O)^0$. We then match them with the ${K}$-types of the genuine (i.e. representations which do not factor to $SO(2n,\mathbb C)$) unipotent representations attached to $\mathcal O$.