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

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.