Principal representations of SO(p,p+1)

TL;DR

This paper studies certain unitary principal series representations of the Lie group SO(p+1,p+1) for odd p, showing their restriction to a subgroup remains irreducible and equivalent to principal series representations of that subgroup.

Contribution

It demonstrates the irreducibility and equivalence of restricted principal series representations of SO(p+1,p+1) to those of SO(p+1,p), extending understanding of their structure.

Findings

Restricted representations remain irreducible.

Restricted representations are equivalent to principal series of the subgroup.

Irreducibility holds under a maximal parabolic subgroup.

Abstract

For p odd, the Lie group SO_0(p+1,p+1) has a family of unitary degenerate principal series representations realized on the space of real (p+1) by (p+1) skew symmetric matrices, similar to the Stein's complementary series for SL(2n,C) or Speh's representation for SL(2n,R). We consider their restriction on the subgroup G= SO(p+1,p) and prove that they are still irreducible and is equivalent to (a unitarization of) the principal series representation of G, and also irreducible under a maximal parabolic subgroup of G.