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

B. Speh () and T. N. Venkataramana ()
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B. Speh: 310 Malott Hall Cornell University, Department of Mathematics
T. N. Venkataramana: Tata Institute for Fundamental Research

A chapter in Representation Theory, Complex Analysis, and Integral Geometry, 2012, pp 231-249 from Springer

Abstract: 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.

Keywords: Complementary series representations; Restriction; Subgroup (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-8176-4817-6_9

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DOI: 10.1007/978-0-8176-4817-6_9

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