3 . STRUCTURE THEORY FOR 677(2,2 ) AND ITSLIEALGEBRA 7

DEFINITION

2.2. Take T ( ^ 0) G GW(7r;rj) and multiplicity one

if-type

(T*,WT*)

of 7T . Here, r* is the contragredient representation

of r. Then we fix a if-equivariant injection i\ WT* — + Hn and consider

the image of

L(V*),

v* G Wr*,

T(tT.(t;*))07) = (i;*,$w,T(»).

Here, (,) is the canonical pairing on WT x WT*. This equality defines

(up to a constant ) a function $niT(g) G C™T(R\G/K). Here,

C?(R\G/K)) = {f:GC-^Vx® WT \ f(rgk) = */(r) 8 r ^ " 1 ) ^ )

\/(r,g,k) eRxGxK}.

We call $7^ a generalized Whittaker function (= GWF for short) for

7r with if-type r*.

3. Structure theory for SU(2,2) and its Lie algebra

3.1. Lie groups and their Lie algebras. Let G denote the spe-

cial unitary group SU(2, 2) realized as

G =

{geSL(4X)\tgi2,2g

= i2,2}

where 72,2 — diag(l, 1, —1, —1) and we denote by tg and g the transpose

and the complex conjugation of g respectively, i.e. realization G\. We

fix a maximal compact subgroup K of G as follows:

K =

ux

u2

u\,v,2 G C/(2), det(wi)det(ix2) = 1

The Lie algebra g of G is expressed as follows:

0

X\ Xs

Xs X2

tXi

= -Xi(i = l,2), X

3

GM

2

(C),

tv(X1 + X2) = 0

The Lie algebra of K is:

' ' X1

x2

t =

es

P =

X*

*xs

es

Then we have the Cartan decomposition: g = t + p. We fix a maximal

abelian subalgebra o of p as follows:

/ 1 \ / 0 \

0 = RHi + RH2 with Hi =

0

1

V

0

,H2

J

0

V

/

The Lie group A = exp(a) is the identity component of a maximal

R-split torus in G.