10 2. GENERALIZED WHITTAKER FUNCTIONS ON 5(7(2,2)
LEMMA 3.2.
X1 =/
0
-
2c2-1Re(7)£_6
+
2c2-1Im(7)£_5,
X2 =Re(
7
)/
0
+ CJ " ^det H^ - Re(72))£_6 + c2E6 + c^lm(-f2)E_5
+ lm(j)(H1-H2),
X3 =Im(
7
)/
0
- cj^det H; + Re(72))£_5 - c2E5 - c^lm^2)E_6
-Re(-y)(Hi-H2).
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 + E-i 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 gai-a2, we have
E5 = ^ {Ad(a-1)X5 + ci ( ^ ) y5 + Re(7)(/fi - #2)} , (12)
^ JAdCo-^Xe - d (^ ) y6 - Im(7)(^ - #2)} . (13)
EQ
=
Then we have the Iwasawa-Cartan type decomposition of g.
LEMMA 3.4.
g =
Ad(a"1)(5u(^)
+ ns) + a + t
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