Representations associated to small nilpotent orbits for complex Spin groups

Abstract

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