Representations associated to small nilpotent orbits for complex Spin groups

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 $G_0 =\operatorname {Spin}(2n,\mathbb {C})$ be the Spin complex group as a real group, and let $K\cong G_0$ 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 $\operatorname {SO}(2n,\mathbb {C})$) unipotent representations attached to $\mathcal {O}$.