10 2. GENERALIZED WHITTAKER FUNCTIONS ON 5(7(2,2)
LEMMA 3.2.
X1 =/
0

2c21Re(7)£_6
+
2c21Im(7)£_5,
X2 =Re(
7
)/
0
+ CJ " ^det H^  Re(72))£_6 + c2E6 + c^lm(f2)E_5
+ lm(j)(H1H2),
X3 =Im(
7
)/
0
 cj^det H; + Re(72))£_5  c2E5  c^lm^2)E_6
Re(y)(HiH2).
Here, I0 = /=Tdiag(l,  1 , 1 , 1) =
\/^T(/i1
+
h2)
e t .
PROOF.
Note that
dip2loda {x) = yx+tx
x
_
t x
y
We readily obtain the assertion. •
Thus we can take {X\, X2, X3} as a basis of su(£) over R and
[Xl5 X2] = 2X3, [X2, X3] = 2(det ^ ) X
1 ?
[X3, Xx] = 2X2.
We put
X5 = X
3
Im(7)X
1
, X6 = X
2
Re(
7
)X
1
, (11)
and
Yi = Ei + Ei e tc (i = 5,6), A  {cxa\ 
c2a22).
We also note that
y5 + V^Ye = 2{e\ +
e2_),
Y5  V^1Y6 = 2(e^ +
e2+).
This implies,
LEMMA
3.3. For the basis of gaia2, we have
E5 = ^ {Ad(a1)X5 + ci ( ^ ) y5 + Re(7)(/fi  #2)} , (12)
^ JAdCo^Xe  d (^ ) y6  Im(7)(^  #2)} . (13)
EQ
=
Then we have the IwasawaCartan type decomposition of g.
LEMMA 3.4.
g =
Ad(a"1)(5u(^)
+ ns) + a + t