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 Poincaré-Dulac [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 a normal 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.

. Invariant conjugacy to a normal form and linearization in Cn
Let X(Cn , 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.
1 1 The vector field zi1 . . .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 -X1 is 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 = do-1 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 X1 of X is invertible .Then: 1.1 .X is invariantly conjugate, possibly formally to a normal form.If X1 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 m = {f E 0c_,o; f (0) = 0} its maximal ideal.For each non-negative integer k denote ,7c-o = 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 Proof. .2.1 .On one hand we 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 -1X-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 X1 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 ) 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.
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 Xl is in the Poincaré domain and in [SI] if Xl 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 Xl .3.1 .If Xl is not resonant, then X is linearizable in a holomorphic invariant way in the following cases : i) Xl 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 .
3.2 .If Xl 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) Xl 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(C2 , 0) with an average morphism of the group action .Let II : X(C2 , 0) --> X(C2 /F, 0) be the morphism of C-vectorial spaces defined by II(X) _ ¡I7¡-1 ~yEr 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 (X1 ) = Xi if and only if Xl = 1\ (Z1, z2) .
Proo£ Suppose Xl is given in its Jordan canonical form .If Xl has different eigenvalues the condition X1-y = yXl implies y is diagonal, therefore F must be diagonal .If Xly = -yX, implies but F is finite then yn = Id and b must be 0.
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] .