Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

126for j = 1, . . ., d. We observe that the collection of 2m + d vector fieldsL k , L k , Υ j span TM . The same holds for the collection L k , L k , Υ j . Wealso have the commutation relations [Υ j , L k ] = 0 and [Υ j , L k ] = 0. Weobserve that Υ γ h(t) = ∂wh(t) γ for all γ ∈ N d . Let α = (β, γ) ∈ N m × N d .By expanding L β Υ γ h(t) using the explicit expressions (5.2), we obtaina polynomial Q β,γ (t, τ, (∂t α′ h(t)) |α ′ |≤|α|), where Q β,γ is a polynomial in itslast variables with coefficients depending on Θ and its partial derivatives.Conversely, since L k | 0 = ∂ zk at the origin, we can invert these formulas, sothere exist polynomials P α in their last variables with coefficients dependingonly on Θ such that(6.11) ∂ α t h(t) = P α (t, τ, (L β′ Υ γ′ h(t)) |β ′ |≤|β|, |γ ′ |≤|γ|).Lemma 6.2. For every l ∈ N, there exists a complex algebraic mappingΠ l with values in C N n,l defined for |t|, |τ| < ρ3 and |J l 0− J l 0Id | < ε whichis relatively polynomial with respect to the higher order jets Ji α with |α| ≥l 0 + 1, i = 1, . . ., n, such that for every local holomorphic self-mappingh ∈ H ρ 2,ρ 1M,κ 0 ,ε, the two conjugate relations(6.12){J l h(t) = Π l (t, τ, J l 0+l¯h(τ)),J l¯h(τ) = Πl (τ, t, J l 0+l h(t)).hold for all (t, τ) ∈ M with |t|, |τ| < ρ 3 .Proof. Applying the derivations L β Υ γ to (6.9), and using the chain rule,we obtain(6.13) L β Υ γ h(t) = Π β,γ (t, τ, J l 0+|β|+|γ|¯h(τ)),where the function Π β,γ (as the function Π) is holomorphic for |t|, |τ| < ρ 3and |J l 0−J l 0Id | < ε and relatively polynomial with respect to the jets Jα i with|α| ≥ l 0 + 1. Applying (6.11), we obtain the function Π l , which completesthe proof.6.3. Substitutions of reflection identities. Let π t (t, τ) := t and π τ (t, τ) :=τ denote the two canonical projections. We write h c (t, τ) := (h(t), ¯h(τ)).We make the following slight abuse of notation: instead of rigorously writingh(π t (t, τ)), we write h(t, τ) = h(t) and ¯h(t, τ) = ¯h(τ).Let x ∈ C ν and let Q(x) = (Q 1 (x), . . .,Q 2n (x)) ∈ C{x} 2n . As themultiple flow of L given by (5.3) does not act on the (z, w) variables, wehave the trivial but important property h(L z1(Q(x))) = h(Q(x)). Moregenerally, for every multi-index α ∈ N n , we have ∂ α t h(L z 1(Q(x))) =∂ α t h(Q(x)). Analogously, we have ∂α τ ¯h(L z1 (Q(x))) = ∂ α τ ¯h(Q(x)). Since