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