Lie Algebraic Approach to z-Functions and its Equations

A general construction of equations satisfied by the components of z-functions is given by considering the tensor products of modules. As an example, the homogeneous basic realization of,'ll is given, leading to NLS equations. O. Introduction Some years ago, Date, Jimbo, Kashiwara, and Miwa [2, 6] showed that solutions of many soliton equations can be considered as group orbits of highest weight vectors V A of L(A). Kac [7] gave a Lie algebraic interpretation for the case that the representation L(A) remains irreducible when considered as ant-module. Here,. is a Heisenberg subalgebra. One can extend this construction to the case where L(A) does not remain irreducible as an ~.-module. Modified equations can be found by considering L(A)| L(A) (A ~/~) instead of L(A) | L(A). 1. Irreducible Highest Weight Modules We start with a given affine Kac-Moody algebra~(A) with a (n + 1) x (n + 1) matrix A, where,(A) has the generators eo, . . . , en and fo, 9 9 fn and a center c. The principal grading ~ is defined by ~i(ei) = -8(J~j) = 1 (i,j = 0 . . . n). We suppose that we have a Heisenberg subalgebra ~. in~(A) with a basis {Pi, qj, c} (i, j > 0) such that [Pi, q j ]=~0 c (i, jegq, i , j>O) . We also suppose that b(pi) = b(q;) > 0. Take an integrable irreducible highest weight module L(A) (over~(A)) with highest weight vector V h, and denote by II the representation of~(A) it affords. One has [I(c) = rn. Id (cf. Lemma 9.3 of [7]). 254 G. POST We define the grading ~L on L(A) by bL(I-I(g)VA) = -(5(g) (g ~ ~ ). Because b(p~) > 0, II(p;) is locally nilpotent on L(A). The highest weight module L(A) splits up as a direct sum of the irreducible a-modules, all isomorphic to each other (see Lemma 14.4 of [7]): L(A)-~ @ R | R = C [ x I , x 2 . . . . ] i e l and (1.1) II(p,) = rn ~ d | I I ( q i ) = x , | II(c) = m. Id . 1 | u s are the vacuum vectors of the irreducible components with eigenvalue m. Theaq(A)-module L(A) carries a unique nondegenerate contravariant Hermitian form H 1 such that HI(VA, VA) = 1 (see Lemma 11.5 of [7]). 2. The z-Functions and Equations To make r-functions, one has to identify a group G associated to~(A). For example, one could take the group generated by exp(tei) and exp (t f ) (i = 0 . . . n, t ~ C). The r-function z(g) is defined by ~(g) = n(g)VA (g~ a ) . (2.1) To fend equations satisfied by z(g), one considers L(A)| This module is completely reducible (w 10.7 of [7]). The module generated by V A | V A is denoted by Lhig h and the submodule L~ig h by Llo w. In general, Llo w is not irreducible. Orthogonality is taken with respect to H defmed by H(V 1 (~) W,, V2(~) W2) = Hi(V,, V2)'H,(W1, W2). It is now essential that z(g) | z(g) ~ Lhigh, from which, by definition, follows H(V| W, z (g ) | z(g)) = 0 (V| ~VELIow). (2.2) This equation will lead to bilinear equations under one more assumption. We assume that o90(q,-) = a~pi (a; ~ C, o~ o anti-linear Cartan involution). (2.3) According to (1.1), we can write v = ZPi (x )u i (vEL(A), we omit | and also z(g) = Z zk(x)u k. For w~ L ( A ) | L(A) we write W = E ei(x(~ ~ aj(x(1))uJ = E ei(x(~ i , j t,J Suppose Hl(Ui, Uj )=e o and Dx= a l , a 2 , . . . 9 ~X 1 OX 2 LIE ALGEBRAIC APPROACH TO z-FUNCTIONS 255 Using (2.3), one finds Hi (~ , Pi(x)ue, ~ Qj(x)ujl= ~ eo.Pi(mDx)Qj(x)lx=o. (2.4) j ] i,J Changing variables ', and 2yj=x ~ ')