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# SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

## construction,

construction, G-invariant. 98 Then, the modular characters Zp, %b, and 7

99 Q(F1) = Cx,~/I(F1) = Cx,~/I(F 0) + k.Cx, k ~" Cx,k/I(F0(x, 0)) = Cx/I(f0) = Q(f0) while the isotropy subgroup G(0 ' ~,) = G, so that the restriction homomorphism reSG(0, ~,) is the identity. Because the pairing is additive on weights, (6.2) sigG(Q(f0) ) = sigG(Q(f0)sd ) Hence (6.1) and (6.2) give the correct result for F 0. To obtain the result for F, we use the topological determinacy theorem in [131] or [D2]. The proof of the theorem proves that the deformation F t = F 0 + t(F - F0) is an equivariantly topologically trivial family of germs deforming F 0 to F such that for any fixed value of t, F t defines an isolated singularity at 0. Since F t has an isolated complete intersection singularity at O, det(dxFt) ¢ 0 on any branch of F t- 1(0) for any t , 0 < t _< 1, in a sufficiently small neighborhood of 0. Thus, sign(det(dxFt) ) does not change along a branch in the topologically trivial deformation. We conclude that Xd is constant, so the above formula is also correct for F; and part 2) of the theorem follows. Lastly, part 3) of theorem 1 will be proven in §7. Proof of Theorem 2 Because the proof of part 1) in this case is closely related to that for the bifurcation character, its proof is postponed until §7. The first part of the above proof for part 2) of theorem 1 applies just as well to the nonweighted homogeneous case and (6.1) gives the result. Part 3) will also be given in §7. Proofs of the Corollaries Corollary 1 For the correct choice of sign for the linear functional, 3) of theorem 1 is correct. For that choice, we let Xbh + denote the characters for the permutation representations on the half branches with ~, > 0 and < 0. Then, 2)~p = Xbh ++Xbh- and 2~b = Xbh +-Xbh- • Adding and subtracting, we conclude Xp + Xb equals ~bh +. [] Corollary 2 i) of the corollary follows because for odd order groups the modular and ordinary characters agree. For ii) we can use theorem 5 of [I)4] once we have shown that the one- dimensional representation Lo~ given there is trivial. However, Lto is generated by to, the generator for toC the module of dualizing differentials. It is a standard fact that in the complete intersection case a representative for to is given by det(dxF0)-!.clX on all branches on

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