14 STEVEN ZELDITCH
t^'hyp
u
Here, J y is a certain transform of j, with
^mx(T)
= 1 + 0(r" ) (VN), as |r|
—» OD ([Z3], § 1(f)); and II* are certain operators depending only on the
s ,m
displayed parameters ([Z3], § 5).
The evaluations (1.2)-(1.2') and (1.3)-(1.3') are given in [Z2]-[Z3] and
will be assumed without proof in this paper. Perhaps, though, it would be
helpful to sketch the main point.
form: L(x,y) = r(x)K(x,y), where
p
helpfu l t o sketc h th e mai n point . First, the kernel L(x,y) of rR,has the
(1.4) K(x,y) = X Kx-Sy).
For j e
CQ(G),
(1-4) is a finite sum and following a standard argument, the
terms may be re-arranged in conjugacy classes. This leads to the kernels
(1.5) K^(x,y) = S *(x" VVy).
'«r7\r
W
Clearly Kr -.(x,x) is V- invariant and of weight -m if a e S Q. So
Lr
T(X,X)
= r(x)Kr *»(x,x) has weight 0, and its integral over T\G is the inner
product «r,Kr -.. The key identity leading to (1.3) is:
(1.6) ,,K{7} = (J7or)yB«a(7)) + (yx+0)Gm^(a(7)).
Here G is another HC transform, like H but involving another special
function. The second term vanishes if a has weight 0, or if it is from the
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