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.
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