THE BASIC INTEGRAL FORMULA
11
Using these formulas in the definition of the Jacobian J f(£,7j) (see appendix) and the
formula (2-8) for the modular function,
A / . ( * , 0 ) A /.(0 , Y*+1) A •• A (0,y,)||
= 2 2 H-R^-uXi A A Rv-i*Xp A jR^-i^L^-i^yife+i A A R^-i^L^-iYqW
- 2A(ri)\\X1 A A Xp A L^-i^Yk+1 A .. . A i ^ - u ^ H
= 2* H^-x.X ! A .. . A L
r
i J
p
A l i
r
i j f c + 1 A .. . A V l * * i l l
(2-19)
= 2 2 ||u! A ••• A uk A -Ufc+i A ••• A vp A wfc+1 A A wq\\
where, to simplify notation, we have set
m L^-^Xi = Ljj-i + Yi 1 i k
V{ L^-i+Xi k + 1 i p
W{ = Lv-imYi k + 1 i q
Also set V L^-iTM^ and W L^-x^TN^. Then from the left invariance of the metric
and the definition of the X^s and IV s it follows
ui,..., Uk is an orthonormal basis of V D W
vjfe+i,..., vp is an orthonormal basis of (V fl W) D V
itffe+i,... ,wq is an orthonormal basis
o f (vnw^nw
Therefore each U{ is orthogonal to each Vj and each Wj whence (Ia = ax a identity matrix)
det
= det
det
j|ui A ••• A MfcAvfc+i A ••• A vp A wk+i A Ag ||2
{v.i,Uj) (vi,Uj) (Wi,Uj)
(ui,Vj) (vi,Vj) (WitWj)
l(Ui,Wj) (Vi,Wj) (Wi,Wj)\
'h 0 0
0 Ip-k {vi,Wj}
L 0 (wjivi) Tq-k
Ip-k (Vi,Wj)
(wi,Vj) Iq-k
= ||v*+i A " . A vv A Wk+x A A wq
Using this in equation (2-19) yields
(2-20) {Jf)(£,v) = 2 * A(?y)||t;*+1 A A vp A wk+1 A . . . A wk\\
Previous Page Next Page