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.