On the Restriction of Representations of SL(2, ℂ) to SL(2, ℝ)

Abstract

We prove that for a certain range of the continuous parameter, the complementary series representation of SL(2,$\mathbb{R}$) is a direct summand of the complementary series representations of SL(2,$\mathbb{C}$). For this, we construct a continuous “geometric restriction map” from the complementary series representations of SL(2,$\mathbb{C}$) to the complementary series representations of SL(2,$\mathbb{R}$). In the second part, we prove that the Steinberg representation σ of SL(2,$\mathbb{R}$) is a direct summand of the restriction of the Steinberg representation π of SL(2,$\mathbb{C}$). We show that σ does not contain any smooth vectors of π.

Mathematics Subject Classification (2010):22D10