Normal forms of invariant vector fields under a finite group action

Publicacions Matemátiques, Vol 37 (1993), 75-82 . NORMAL FORMS OF INVARIANT VECTOR FIELDS UNDER A FINITE GROUP ACTION Abstract FEDERICO SÁNCHEZ-BRINGAS Let I' be a finite subgroup of GL(n, (C) . This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of ((Cn, 0) . We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2 . Introduction The goal of this paper is to show that the classic theorems of PoincaréDulac [DU] and Siegel [SI] of conjugacy to a normal form and linearization of germs of holomorphic vector fields at 0 E Cn hold for the quotient space Cn/I', where I' is a finite subgroup of GL(n, C) . In this situation we consider the germs of holomorphic vector fields and the germs of conjugating diffeomorphism of Cl invariant by the action of the subgroup . It is well known that Cn/I' has the structure of an algebraic variety and furthermore any variety which is the quotient of a finite group of local diffeomorphisms of Cn is of this form (in a specific system of coordinates) [CA], then we obtain here results for conjugacies to normal forms and linearizations of germs of holomorphic vector fields in this kind of algebraic varieties . In a different context, like bifurcation theory, sometimes conjugacy to anormal form of germs of holomorphic vector fields which preserves symmetries are needed, this results can also be applied . In the first section we prove the main theorem using the algebraic approach developed in [CH] . In the second section we analyse carefully the case C'/I` and we give a description of normal forms . Finally we wish to thank Xavier Gomez-Mont for líis helpful comments and remarks concerning this work . 7 6  F . SÁNCHEZ-BRINCAS 1 . Invariant conjugacy to a normal form and linearization in Cn Let X(C n , 0) be the space of germs of holomorphic vector fields at 0 E Cn vanishing at the origin . Let F be a finite subgroup of GL(n, C) which acts naturally on X(Cn, 0), we say that X is invariant if it is invariant by this action, namely if for all ,y E F, d " y-1(X(1'(z)) = X(z) . Let X(C n /F, 0) be the subspace of invariant elements of X(Cn, 0) . Given X E X(Cn,0) denote by X 1 its linear part, dX(0) and suppose it belongs to GL(n, C) . Let S be the semisimple part of X1, we say X is a normal form if LSX = 0, where L s is the Lie derivative of S . X1 is said to be resonant if its eigenvalues satisfy a relation (resonante) like this : A v =  M i ñ2, u = 1, . . . , n, (m1, . . . , m,,) E Nn,  m i i 2 . 1  1 The vector field zi 1 . . . zm ~ aáu , u = 1, . . . , n is the monomial vector field associated to this resonante . Suppose that X E X(C', 0) is a normal form and the coordinates of Cn are given by a basis of eigenvectors of the semisimple part S of X1 . Then condition LSX = 0 implies that X - X 1is a sum of resonant monomial vector fields . Let Diff (Cn, 0) be the group of germs of holomorphic diffeormorphisms which fix the origin of Cn, F acts by conjugation here . We say that 0 E Diff(Cn,0) is invariant if it is invariant by this action, namely if for all ,y E F, ,y-10 y = 0 . Denote by Diff(Cn/F, 0) the group of invariant elements of Dif(Cn, 0) . For any X, Y E X(C n , 0) we say X is conjugate to Y if there is a E Diff(Cn, 0) such that 0* X = Y, where 0* X = do1 XO . When Y is the linear part of X we say X is linearizable . If X and Y are invariant, the conjugacy (linearization) is called invariant . Theorem  1 . Let  F  be  a finite  subgroup  of GL(n, C)  and X E X(C n /I', 0) . Suppose the linear part X 1 of X is invertible . Then : 1 .1 . X is invariantly conjugate, possibly formally to a normal form . If X 1 is non-resonant then this conjugacy is an invariant linearization . 1 .2 . If X is holomorphically conjugate to a normal form, then it can be conjugate in an invariant holomorphic way . The proof of this theorem is a consequence of the following lemma . Let OC .,o be the algebra of germs of holomorphic functions at 0 E C" , and NORMAL FORMS OF INVARIANT VECTOR FIELDS  77 m = {f E 0c_,o ; f (0) = 0} its maximal ideal . For each non-negative integer k denote ,7co = 0cn,o/m k the algebra of finite dimension of k-jets of holomorphic functions . The element X E X(Cn, 0) defines a derivation X* of Oc~,o, X* f = LX f and in a natural way the k-jet of X determines a derivation X * of ,7c_ ,o, then X * has a canonical decomposition : Xk = Sk + Nk , where Sk is the semisimple part and Nk is the nilpotent part . A remarkable fact proved in [CH] is that S* is a derivation . In a similar way, we denote by Diffk(Cn, 0) the group of k-jets of germs of holomorphic diffeomorphisms of Cn . The definition of conjugacy to a normal form (linearization) is extended in a natural way to the space of k-jets of germs of vector fields, Xk(Cn,o) . Lemma 2 . Let F be a finite subgroup of GL(n, C), k a non-negative integer and Xk E X k (C n /I', 0) then 2 .1 .  The semisimple part S* of X * is invariant . 2 .2 . S* is invariant linearizable if X 1 is invertible . Proof . . 2 .1 . On one handwe have the following fact [Hu] : Let V be a C-vector space of finite dimension and T an endomorphism of V . Then the semisimple part of T has a polynomial expression in T, p(T) with coefficients in C . On the other hand, as X is invariant and y E F is linear we have d-y - 'Xy= y -1 X-y = X then for any nonnegative integer k, y - 1 Xky =X* and -y-1X% o . . . oXky = X* o . . . oX* then any polynomial expression in X* with coefiicients in C is invariant . 2 .2 .  Let Ok E Diffk(Cn, 0) be the Poincaré-Dulac diffeomorphism which exists because X 1 is invertible . Ok is tangent ((k - 1)-order) to the identity diffeomorphism, and linearizes the semisimple part of Xk . Define the average ~k = 1 FI -1 1 :,, Er , y -1 Oky . Ok is invariant and tangent ((k - 1)-order) to the identity diffeomorphism . Besides y-1 0ky I (S*) = ¡FI-1 57 (y -1 0k 1 y)*S*(y -1 0ky) 7EF  yEr = IFI -1  ySiy_ 1 -yEF where Si is the linear part of Sk which is invariant because of Sk . 7 8  F . SÁNCHEZ-BRINGAS Proof of the theorem : 1 .1 . Ls,Xk = 0 if and only if 0 = (Ls,Xk)* = Si X¡* -Xk Si , then the canonical decomposition of X¡ * implies that conjugating to a normal form in xk(C , ,0) is equivalent to linearizing the semisimple part Sk* . Now let 4 be like in lemma 2 . Remark that if we write 010 . . . 0 w2 = Id +02 + . . . + 01 + . . . then 01+1 0 01 0 . . . o 02 = (Id+02+ . . . +~~+~ 1 + 1 + .. . )-}-~t+1(Id+ . . . )+ . . . =Id+02+ . . .+01+ .. . so this two compositions llave the same 1-jet and Ok o . . . o 02 conjugates invariantly Xk to a normal form because as we showed, this diffeomorphism linearizes S¡ . Finally the limit limk -w (& o . . . o w2) defines a diffeomorphism 0, eventually formal which conjugates invariantly X to a normal form . 1 .2 . Let 0 be the holomorphic conjugacy (in any system of coordinates) then a similar argument as in 1 implies that ¡F¡ -1 E~, EI , " y - 'Oy E Diff(C', 0) conjugates X holomorphically and invariantly to the respective normal form . 2 . Description in C 2 : Invariant normal forms Let Xl be a linear vector field in C 2 with eigenvalues >11, 1\2 . Choose a base of C 2 , {el, e2} of eigenvectors of S, the semisimple part of Xl . We say that X l belongs to the Poincaré domain if 0 is not in the segment [Al, A2] . Otherwise we say Xl belongs to the Siegel domain . The eigenvalues A1, A2 are of type (C, v), C, v >_ 0 if : ¡w - mlal - MA21> Qml1 + Im2j) - " for all (ml, m2) E (N')*, u = 1, 2 . Before applying theorem 1 in this context we point out that conditions to X be conjugate holomorphically to a normal form llave been established in [DU] if X l is in the Poincaré domainand in [SI] if X l belongs to the Siegel domain . Theorem 3 . Let Xl be the linear part of X E x(Cn/F, 0) . Let A1, A2 be the eigenvalues of X l . 3 .1 . If X l is not resonant, then X is linearizable in a holomorphic invariant way in the following cases : i) X l belongs to the Poincaré domain . ii) Xl belongs to the Siegel domain and Al, A2 are of type (C, v) for some C, v > 0 . NORMAL FORMS OF INVARIANT VECTOR FIELDS  79 3 .2 . If X l is resonant, then X is conjugate in a holomorphic invariant way to a normal form in the following cases : i) Xl belongs to the Poáncaré domain . ii) X l belongs to the Siegel domain and the normal form is colinear to Xl . Remark . There exist cases where the invariant conjugacy to a normal form is only formal . For example if F is a diagonal group (Le . each of its elements are diagonal) we are going to show there are normal ffoms with linear part in the Siegel domain which do not verify condition 2,ii) . In this case the conjugating diffeomorphism 0 may be divergent because one of its coordinate functions can have coefficients which grow like the Euler function, [Br] : 00 We express the invariance condition in X(C 2 , 0) with an average morphism of the group action . Let II : X(C 2 , 0) --> X(C 2 /F, 0) be the morphism of C-vectorial spaces defined by II(X) _ ¡I7¡-1 ~ y Er y*X . Then X is invariant if and only if fi(X) = X . There are two different cases for the family of finite subgroups of GL(2, C) : i) If F is diagonal, the monomial vector fields are eigenvectors of II and II(X) = 0 if X is not invariant . ii) If F is not diagonalizable the eigenvectors of II are not monomials and II does not vanish monomials . Let us regard first the case of diagonalizable groups . Proposition 4 . If F is not diagonalizable and Xl is a linear vector field, then II (X 1) = Xi if and only if Xl = 1\ (Z1, z2) . Proo£ Suppose X l is given in its Jordan canonical form . If Xl has different eigenvalues the condition X 1 -y = yXl implies y is diagonal, therefore F must be diagonal . If X l y = -yX, implies but F is finite then yn = Id and b must be 0 . 8 0 F . SÁNCHEZ-BRINGAS Remark . This proposition implies that for non diagonalizable groups our theorem is a linearizing theorem illustrated by the following example : Let F be the binary dihedral group generated by _ ra0l and _ (0  1l 10 -i/  11 Ól The vector field X(zl, z2) = (zl, z2) + (zi ; z2) is not a multiple of the radial vector field' which belongs to x(C 2 /F, 0) . Suppose now the group is diagónal . In order to simplify the description of invariant normal forms we will Suppose F is cyclic and generated by e (27rli v '-1)/ni 0 e(27rl2v~---1-)/n2 n  , E N, 1,, E Z, (l ti ,, n ti ,) = 1, u = 1, 2 . If X(zl, z2) = Eu=1,2(E¡~_o CI'j .ziz2)eu, the equivariance condition is imposed independently for monomial vector fields . The next proposition describes invariant vector fields . Proposition 5 . Let n be the least common multiple of n1, n2 and rlu = nlu /n u . The monomial vector field zu(zúzv)eu , u = 1, 2, u =,A v, i >_ -1, j > 0', i + j >_ 0 is invariant 'if and only if rlui = -?7 vj (mod n) . Proof . . Suppose belongs to F then II(z1 2e u ) = ( 1 / n Z :7Er wl,yi72)zizjeu, therefore zizieu is invariant if and only if n-1 r7EP 7u 1 i7z = 1 .  If u = 1, r-^YEr ^Yi -1 Y2 - rk=1 e2~rv~k((li/ni)(a-1)+(12/n2) .Í)  This sum is n if (n/n1)lli + (n/n2)l2j - (n/n1)l1(modn) and vanishes otherwise . Similarly for u = 2 . When X 1 belongs to the Poincaré domain and its eigenvalues are resonant, then the origin, A1 and A2 are colinear, besides 0 1 [A1, a2], therefore there is only one possible type of resonance : Au = mvw, u =,/= v . When X 1 belongs to the Siegel domain the resonance Au = muAu + m vA, generates an infinite family of resonantes of type : Au = k((mu - 1)Au + mvw) + ñuñv = k((mu - 1)Au + m ñv) + w, keN,u7~v . Finally let us make the - following classification : NORMAL FORMS OF INVARIANT VECTOR FIELDS  8 1 Proposition 6 . Let I' be a finite diagonal cyclic subgroup of GL(2, C) . The invariant normal forms are : 1 . If X1 belongs to the Poincaré domain . X (z1, z2) = X1( 2 1, z2) + a¡zveu where 17 v i - r7 u (modn), u qÉ v and u, v = 1, 2 . 2 . If X1 belongs to the Siegel domain . X(z1, z2) = X1(z1, z2) + Cz1 (É akz1iz2j/ , z2 (E bkz1 i z2 j k=1  k=1 where r71i - -772 j (mod n) or k - 0(mod n) and i, j are like in proposition 5 . For all cases if l1/n1 - l2/n2 0 Z, then X1 is diagonal . This section can be applied to obtain conjugacies to normal forms in surfaces of type C 2 /P where I' is a finite subgroup of SU(2) . These surfaces are embedded in C 3 with an isolated singularity at the origin [KLE] . If I' is diagonal, then it is cyclic and generated by e 27r V I'--- 1 /n  0 0 e -27rv - -l/n Proposition 6 applies in this case : 971 = -7 72 = 1, l1/n1 - l2/n2 = 2/n1Zifn7~ 2 . When I' is non-diagonalizable, we have the groups which are the inverse image of the covering surjection p : SU(2) ---> SO(3) of the groups of index 2 (preserving orientation) of triangular spherical groups [MIL] . References [BR] BRJUNO, A . D ., Analytic forms of Differential Equations, 7yans . MoscowMath . Soc . 25 (1971), 131-282 . [CA] CARTAN, H ., "Quotient d'un espace analytique par un groupe d'automorphismes," Algebraic Geometry and Topology, Princeton U .P ., 1957 . [CHI CHAPERON, M ., "Invariant manifolds and a preparation lemma for local holomorphic flows and actions," Holomorphic dynamics, Proceedings of Springer Verlag, Lecture Notes in Math . 1345, 1986 . 8 2  F . SÁNCHEZ-BRINCAS [DU] DULAC, H ., Solutions d'un systeme d'equations differentielle's dans le voisinage des valeurs singuliéres, Bull . Soc . Math . France 40 (1904) . [HU] HUMPHREYS, J ., "Introduction to Lie algebras and representation theory," Springer Verlag . [KLE] KLEIN, F ., "Lectores on the Icosahedron and the solution of the equations of fifth degree," Teubner 1884, Dover, 1956 . [MIL] MILNOR, J ., "On the 3-dimensional Brieskorn manifolds M(p, q, r)," Ann . of Math . Studies 84, Princeton U .P ., 1975 . [SI] SIEGEL, C . L ., "ber die normalform analytischer differentialgleinchungen in der Nhe einer gleichgewichtslosung," Nchr . Akad Wiss ., Gottingen Math-Phys . kl, Math .-PhisChem . Abt, 1952, pp . 21-30 . Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria México 04510 D .F . MÉXICO Primera versió rebuda el 30 de Gener de 1992, darrera versió rebuda el 13 d'Octubre de 1992