Singularities of Intertwining Operators and Decompositions of Principal Series Representations

Lior Silberman, Taeuk Nam, Avner Segal

Abstract:

In this paper, we show that, under certain assumptions, a parabolic induction IndGBλ from the Borel subgroup B of a (real or p-adic) reductive group G decomposes into a direct sum of the form:
IndGBλ=(IndGPStM⊗χ0)⊕(IndGP1M⊗χ0),
where P is a parabolic subgroup of G with Levi subgroup M of semi-simple rank 1, 1M is the trivial representation of M, StM is the Steinberg representation of M and χ0 is a certain character of M. We construct examples of this phenomenon for all simply-connected simple groups of rank at least 2.