5 years ago

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

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

4 Z2-SYMMEZ Y 210 We

4 Z2-SYMMEZ Y 210 We consider corank 1 Lagrange projections from Z2-invariant Lagrange submanifolds of T*IR n, where Z 2 = {1,~c} acts on V = Rn by :- ~:- (Xl ..... Xr, Yl ...... Ys) = (-Xl,--..,-Xr, Yl,-'--,Ys) ( r + s - n ). We also assume that Z 2 acts nontrivially on W* = R; the case of a trivial action is dealt with by Remark 2.5. The main results are Corollary 4.4, which states that if r < s there are generic E 2 equivariant Lagrange projections which are not infinitesimally Z 2 stable, and Theorem 4.4, which includes a complete classification of infinitesimally stable Z 2 equivariant Lagrange projections when n

211 where q%(x,y) = Z Vbc(X'y)xc and the %a and the Vbe are Zz-invariant C=I functions of x and y. Moreover since ~2(k+l) is simple, the restriction of F to W*~V G will be 5~ G versal. Hence the mapping X will be a submersion and we can choose coordinates YI ..... Ys so that Xa = Ya. Condition 1.2 implies that (VII(0) ...... Vlr(0) )~ 0. By a linear change of the coordinates Xl,.;.,x r we may suppose that (V1 1(0) ...... Vlr(0) ) = (1,0 ..... 0). Finally, by redefining x 1 we may take V1 l(x,Y) to be identically 1. ¢ We now derive necessary and sufficient conditions on the 222-invariant mapping : V -* M(k,r) for (4.1) to be infinitesimally stable, the target of V being the space of k x r matrices, Proposition 4.2 F is infinitesimally stable if and only if the restriction of V obtained by putting x i=0=yjfor i--1 ...... r and j= 1 ...... k:- : IR s-k ~ M(k,r) )s (Yj j=k+l I ~ ~(0,0,yk+l ...... Ys), is transversal at y = 0 to the orbits of the natural action of GL(r) on M(k,r). Proof By Proposition 2.7, F is infinitesimally stable if and only if the 22 2 equivariant map germ • (x,y) -~ (Yl ....... Yk, q°l(x,Y) ......... q)k(x,y)) is 5~Z2- stable. The result above follows from the characterisation of 5~Z2- stable germs in [ R1 §4 and R2 ]. Remark 4.3 This transversality criterion is equivalent to the condition that the following matrix has rank kr, the maximum possible:- ¢

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