Representations associated to small nilpotent orbits for real spin groups

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 G₀ = 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 G₀. We consider the nilpotent orbit O_c parametrized by [3 2^2k 1²n^{− 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 ψ 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.