14 1. THE CLASSICAL THEORY: PART I For n = 1 we define the norm by sup y0 ∞ −∞ |fψ(x + iy)|2dx. The spaces D− n are described analogously using the lower half plane. Fact ([Kn2]). The D± n for n 2 are the discrete series representations of SL2(R). For n = 1, D± 1 are the limits of discrete series. The terminology arises from the fact that in the spectral decomposition of L2(SL2(R)) the D± n for n 2 occur discretely. There is an important duality between the orbits of SL2(R) and of SO(2, C) acting on P1. Anticipating terminology to be used later in these lectures we set • P1 = flag variety SL2(C)/B where B is the Borel subgroup fixing i = [ i 1 ] • SL2(R) = real form of SL2(C) relative to the conjugation A → A • SO(2) = maximal compact subgroup of SL2(R) (in this case it is SL2(R)∩ B) • H = flag domain SL2(R)/ SO(2) • SO(2, C) = complexification of SO(2). We note that SO(2, C) ∼ C∗. Matsuki duality is a one-to-one correspondence of the sets {SL2(R)-orbits in P1} ↔ {SO(2, C)-orbits in P1} that reverses the relation “in the closure of.” The orbit structures in this case are H✾✾ ✾✾✾ ✾✾✾ H ☎☎☎ ☎☎☎ ☎☎ open SL2(R) orbits R ∪ {∞} closed SL2(R) orbit P1\{i, −i} ☎ ☎ ☎☎ ☎ ☎ ☎ ❀ ❀❀ ❀❀ ❀ ❀ open SO(2, C) orbit i −i closed SO(2, C) orbits The lines mean “in the closure of.”5 The correspondence in Matsuki duality is H ↔ i H ↔ −i R ∪ {0} ↔ P1\{i, −i}. Matsuki duality arises in the context of representation theory as follows: A Harish-Chandra module is a representation space W for sl2(C) and for SO(2, C) that satisfies certain conditions (to be explained in Lecture 5). A Zuckerman module is, 5 Matsuki duality for flag varieties is discussed in [FHW] and in [Sch3] where its connection to representation is taken up.

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