Corners and fundamental corners for the groups Spin(n,1)

TL;DR

This paper investigates the structure of irreducible non-elementary representations of Spin(n,1), revealing bijections with the dual of the maximal compact subgroup, extending known results for similar groups.

Contribution

It extends the understanding of corners and fundamental corners for Spin(n,1), establishing new bijections between non-elementary parts of the duals for even and odd n.

Findings

For even n, a bijection between non-elementary duals and the dual of the maximal compact subgroup.

For odd n, a bijection exists between the non-elementary dual and a proper subset of the compact dual.

Results generalize previous work on SU(n,1) to the Spin(n,1) groups.

Abstract

We study corners and fundamental corners of the irreducible representations of the groups G=Spin(n,1) that are not elementary, i.e. that are equivalent to subquotients of reducible nonunitary principal series representations. For even n we obtain results in a way analogous to the results in [10] for the groups SU(n,1). Especially, we again get a bijection between the nonelementary part $\hat{G}^0$ of the unitary dual $\hat{G}$ and the unitary dual $\hat{K}.$ In the case of odd n we get a bijection between $\hat{G}^0$ and a tru subset of $\hat{K}.$