5 years ago

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

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

202 For any G-invariant

202 For any G-invariant function germ F : W* ~ V, 0 ~ N we define the tangent space :- + TG(F) = L(~G "F G The first term is the ideal in ~p,q G G the ~q-submodule of ~p,q , {~I.F .......... 13s.F,1). G G ~p,q {(Zl.F .......... Otr.F) + Eq {131.F .......... 13s.F, 1). generated by {Otl.F .......... CZr.F}. The second term is G thought of as an [q-module, generated by Infinitesimal Lagrange stability for invariant Morse families is defined in the usual way :- Definition 2.3 A G-invariant function germ G TG(F ) = ~p,q. F : W* ~V ~ IR + is infinitesimally 5~ G stable if Standard singularity theory ideas would show that infinitesimal stability of Morse families is equivalent to stability, and hence to equivariant symplectic stability of invariant Lagrange submanifolds. However this is not pursued in this paper. Recall that in the absence of a group action the infinitesimal 5~ + stability of an unfolding is equivalent to its 5~ versality, where 5~ is the group of germs of diffeomorphisms of W*. There is no analogue of this result in the equivariant case, though we do have the following result. Let Ep denote the space of function germs on G W*, Ep the space of G-invariant germs and mp and m their respective ideals of G germs vanishing at the origin. Let J(f) denote the Jacobian ideal of f~ Ep; if f~ Ep then J(f) is invariant under the natural action of G on ~p. Proposition 2.4 + If F is an infinitesimally 2G-stable Morse family, then the restriction FIW,~VG is an

G $~G-versal unfolding of F(.,0) in mp. Proof If F is infinitesimally stable the mapping:- 203 ) G ~pq induced by inclusion, is surjective. This implies that the mapping:- G G G ~G{ocI.F ........ O~r.F} + (mqEpq) f~Eq {~I.F ....... ~s.F, 1} (mqEpq) is surjective. If ql ..... qa denote coordinates on V G and f = F(.,0), then this can be written as:- G ~.{ ~F (q=0)}i=l,..., a = mp Oqi j(f)G ' which is the infinitesimal criterion for ~G versality. Remark 2.5 If K is the kernel of the action of G on W* then infinitesimal ~gG-versality is the same as infinitesimal ~G/K-versality. In particular, if G acts trivially on W*, then the restriction of an infinitesimal NG-versal Morse family, F, to W*~V G is versal in the + usual sense. It follows from the work of Slodowy [S] that F must be 5~G-equivalent to a trivial extension of HW,~VG. An averaging argument shows that TG(F) = (T(F)) G, the fixed point subspace for the induced action of G on the full 5~ + tangent space. It follows that if an unfolding is + infinitesimally ~+ stable, in other words versal, then it is also infinitesimally ~G- stable. The examples in §4 show that the converse is not true. Slodowy IS] has introduced another notion of versality for invariant functions, for which we reserve the description G-versaL This turns out to be particularly useful when G is a finite group. Recall first, from [Rl,§4], that finite ~G determinacy holds in ) G

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