192

J . - M . BISMUT AND W. ZHANG

GAFA

So for g € G

log ( |tdetPoo , r||L^HF^ - ^G ) ^det ( C - ( H^« , F ) , G ) ) ( ^ ) = (6.62)

2^ log yi det oo,Tlldet-i(HomG(W,F!^'^b®W)<8)det(HomG(VV^,C«(\y«,F))!8)W)^

= Trs[9log{P:,^TPoo,T)]

het - '4EomG { W , ¥l^'^^ ) i^W ) met { EomG { W , C * { W^ , F ) ) ^W ) )

x { W ) { g )

Tk { W )

Prom (6.60)-(6.62), we get (6.57). d

e ) Proof of Theorem 5.5.

Clearly the operators Р|^' yP^ commute with G. Similarly, by struction, Jt commute with G. So e^, ет, Роо.т^т also commute with G. By Proposition 6.16 and Theorem 6.17, for T > 0 large enough, g £ G^

Ъгз [iV5log(DjJ"'^')](5) + log (I \ff^H4M,F),G),T)\9) (6-63)

+ Tts [ölog(P4,rPoo,T)] = log (II |1а'^Гя.(м,л,с))'(5) •

Using the multiplicativity of the determinant, we get for T > 0 large enough,

Tts [^log(Fi,7.Poo,r)] =Tr, [fflog((Poo,TeT)*Poc,reT)]-Tr, [^log(e^eT)] .

( 6 . 64 ) By Theorem 6.9, there is с > 0 such that as Г —^ +oo

It , [fflog(e^er)] = OCe-'^^) . (6.65)

Since Poo,T^T aJid e-^'^(^)^"'"/^ commute with G, the operator C?{e~'^^) appearing in (6.38) also commutes with G. By Theorem 6.11,

{ Роо , тетГРоо , тет= { 1 + 0{е-^^)У (^)''""^%2Т^(1 + 0(e-^)) .

( 6 . 66 ) Using again'the multiplicativity of the determinant and the fact that all the operators appearing in (6.66) commute with G, we get

Tr , [ 3log { ( Poo , Ter ) * Poo . reT ) ] =1^, öbg ((^)''~"^%'^^)]+C?(e-'=^).

( 6 . 67 )