14 1. THE CLASSICAL THEORY: PART I
For n = 1 we define the norm by
sup
y0

−∞
|fψ(x +
iy)|2dx.
The spaces Dn are described analogously using the lower half plane.
Fact ([Kn2]). The Dn ± for n 2 are the discrete series representations of
SL2(R). For n = 1, D1
±
are the limits of discrete series.
The terminology arises from the fact that in the spectral decomposition of
L2(SL2(R)) the Dn ± 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,
5Matsuki
duality for flag varieties is discussed in [FHW] and in [Sch3] where its connection
to representation is taken up.
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