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

302At the end, we allow to suppress the projection π z : this slight abuse of notationwill lighten slightly the writting of further formulas. More generally,for λ ∈ N, a 1 ∈ K p , x 1 ∈ K n :(11.29)J λ z ϕ( L ∗ a 1(Q(θ)) ) ≡ J λ z ϕ( Q(θ) )J λ c h ( L x1 (Q(θ)) ) ≡ J λ c h ( Q(θ) ) .As a consequence, for 2k even and for 2k + 1 odd, we have the followingfour cancellation relations, useful below (we drop π z and π c after J λ z ϕ andafter J λ c h):and(11.30)⎧Jz λ ϕ( Γ 2k ([xa] 2k ) ) ≡ Jz λ ϕ( Γ 2k−1 ([xa] 2k−1 ) ) ,⎪⎨ Jc λ h ( Γ ∗ 2k([ax] 2k ) ) ≡ Jc λ h ( Γ ∗ 2k−1([ax] 2k−1 ) ) ,Jz λ ⎪⎩ϕ( Γ ∗ 2k+1 ([ax] 2k+1) ) ≡ Jz λ ϕ( Γ ∗ 2k ([ax] 2k) ) ,Jc λ h( Γ 2k+1 ([xa] 2k+1 ) ) ≡ Jc λ h( Γ 2k ([xa] 2k ) ) .We are now in position to state and to prove the second main technicalproposition.Proposition 11.31. For every even chain-length 2k and for every jet-heightλ, there exist two local K-analytic maps, Φ λ 2k valued in K(n+m)Cλ n+m+λ , andH λ 2k valued in K(p+m)Cλ p+m+λ such that:(11.32){Jλz ϕ ( ( )) (Γ ∗ 2k [ax]2k ≡ Φλ2k [ax]2k , J k(κ+κ∗ )+λz ϕ(0) ) andJ λ c h ( Γ 2k([xa]2k))≡ Hλ2k([xa]2k , J k(κ+κ∗ )+λc ϕ(0) ) .Similarly, for every odd chain length 2k + 1 and for every jet eight λ, thereexist two local K-analytic maps, Φ λ 2k+1 valued in K(n+m)Cλ n+m+λ and Hλ2k+1valued in K (p+m)Cλ p+m+λ , such that:(11.33) {Jλz ϕ ( ( )) (Γ 2k+1 [xa]2k+1 ≡ Φλ2k+1 [xa]2k+1 , J kκ+(k+1)κ∗ +λc h(0) ) ,Jc λ h ( ( )) (Γ ∗ 2k+1 [ax]2k+1 ≡ Hλ2k+1 [ax]2k+1 , J (k+1)κ+kκ∗ +λz ϕ(0) ) .These maps depend only on Π, Π ∗ , Π ′ , Π ′∗ .Proof. For 2k + 1 = 1, we replace (z, c) by Γ 1 ([xa] 1 ) in the first lineof (11.27) and by Γ ∗ 1 ([ax] 1) in the second line. Taking crucially account