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

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

200 2 SYMPLECTIC

200 2 SYMPLECTIC EQUIVALENCE In this section we introduce the equivalence relation we shall use to classify equivariant Lagrange projections. In the absence of a group action the theory described reduces to the "classical" one, as described in [AGV], for example. We keep the same notation as in §1. Definition 2.1 Two G-invariant Lagrange submanifold germs Lj,0 c T* V,0 (j = 1,2) are symplectically equivalent if there exist germs of a G-equivariant symplectomorphism : T'V,0 ~ T'V,0 and a G-equivariant diffeomorphism (p : V,0 ~ V,0 such that :- (i) the following diagram commutes:- T'V,0 V,0 (ii) and ~(L1) c L 2. O q~ T*V,O Let W be a representation of G which is isomorphic to a G-invariant subspace of V. The following result is proved exactly as in the non-equivariant case, see [AGV]. Proposition 2.2 Two G-invariant Morse families Fj : W*~V,0 ~ IR (j = 1,2), generate symplectically equivalent Lagrange submanifolds if and only if there is a G-equivariant diffeomorphism germ ~F : W* ~V,0 ~ W* ~V,0, a G-equivariant diffeomorphism germ V : V,0 ~ V,0 and a G-invariant function germ o~ : V,0 ~ ~1, such that :- (i) the following diagram commutes: V,0

(W*~V,0) (v,o) (where n2 is the natural projection ), 201 (w ~v,0) fv,o) (ii) and Fl(p,q) = F2(W(p,q)) + offq) with p a W*, q a V. This is essentially the same as the equivalence of equivariant unfoldings defined by Slodowy IS], and in the absence of a group action is the ~+-equivalence of [AGV]. We + will denote the group of equivariant equivalences by 5~ G and if ~e conditions of the + proposition hold we say that F 1 and F 2 are 5~Gequivalent.. G Next we describe the tangent space for this equivalence relation. Let Ep,q (respectively G ~q ) denote the ring of germs of G-invariant smooth functions on W*@V (respectively G G V). Let {oq ........ CXr) denote a generating set for the ~p,q -module, O~, consisting of germs of G-equivariant vector fields along the projection ~ : W*@V-~ W*. These are k G-equivariant vector fields of the form ~ ai(p,q) ~ . i= I OPi 2 Let {[~1 ...... ~s} denote a G G generating set for the Eq -module, ®q, of germs of G-equivariant vector fields on 4- G c (V,O). We may regard the direct sum ~,5~ G = O~ @ O @ Eq as the Lie algebra of the 4- group ~G' the three summands consisting of infinitesimal coordinate changes corresponding to W,~g and a (respectively) in Proposition 2.2.

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