THE W1+∞(gls)–SYMMETRIES OF THE S–COMPONENT KP HIERARCHY

Adler, Shiota and van Moerbeke obtained for the KP and Toda lattice hierarchies a formula which translates the action of the vertex operator on tau–functions to an action of a vertex operator of pseudo-differential operators on wave functions. This relates the additional symmetries of the KP and Toda lattice hierarchy to the W1+∞–, respectivelyW1+∞×W1+∞– algebra symmeties. In this paper we generalize the results to the s–component KP hierarchy. The vertex operators generate the algebra W1+∞(gls), the matrix version of W1+∞. Since the Toda lattice hierarchy is equivalent to the 2–component KP hierarchy, the results of this paper uncover in that particular case a much richer structure than the one obtained by Adler, Shiota and van Moerbeke. §0. Introduction. The KP hierarchy is the set of deformation equations ∂L ∂tk = [(L)+, L], for the first order pseudo-differential operator L ≡ L(x, t) = ∂ + u1(x, t)∂ −1 + u2(x, t)∂ −2 + · · · , here ∂ = ∂ ∂x and t = (t1, t2, . . . ). It is well–known that L dresses as L = P∂P −1 with P ≡ P (τ, x, t) = 1 + a1(x, t)∂ −1 + a2(x, t)∂ −2 + · · · = τ(x, t− [∂]) τ(x, t) , where τ is the famous τ–function, introduced by the Kyoto group [DJKM1-3] and [z] = (z, z 2 2 , z 3 , . . . ). The research of Johan van de Leur is financially supported by the “Stichting Fundamenteel Onderzoek der Materie (F.O.M.)”. E-mail: [email protected] Typeset by AMS-TEX 1 2 JOHAN VAN DE LEUR The wave or Baker–Akhiezer function Ψ ≡ Ψ(τ, x, t, z) = W (τ, x, t, ∂)e, where W ≡ W (τ, x, t, z) = P (τ, x, t)e with ξ(t) = ∞ ∑ k=1 tk∂ k is an eigenfunction of L, viz., LΨ = zΨ and ∂Ψ ∂tk = (L)+Ψ. From this point of view, the introduction by Orlov and Schulman [OS] of another pseudodifferential operator M ≡ M(x, t) = WxW which action on Ψ amounts to MΨ = ∂Ψ ∂z is rather natural. Recently, Adler, Shiota and van Moerbeke [ASV1], [ASV2] proved a conjecture of Orlov and Schulman, viz., that there exists a relation between (M L)− acting on Ψ and the generators W (l+1) k ∼ −t ( ∂ ∂t ) of the W1+∞–algebra acting on the τ–function. More explicitly, let Y (τ, y, w) = ∞ ∑