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Be elements of R. (2n-r-l)! (r) E M 2k (r) for all k ~ 1; fn-r(z) where fk E M2k(r) (Vk ~ 1). Proof. One checks that each of the properties in question is equivalent to the transformation law (3). 9). Property (4) can be checked the same way or else we can note that by a simple binomial coefficient identity it is equivalent to (5) if we define Ik to be 2k - 1 times the sum in (4). 0 We can restate the result of Proposition 2 in the following way. Denote by M(r)+ = TIn>O M 2n (r) the space of sequences of modular forms of positive weights, with the trivial filtration by the subspaces M(r)k = TIn>k M 2n (r), with successive quotients M(r)k/M(r)k-l = M 2k(r).

Almost immediately they were generalized to matrix equations and formed "multi-component" KdV's and KP. All the above mentioned hierarchies are generated by linear differential (KdV) or pseudo-differential (KP) operators of arbitrary orders. Equations of another type are generated by matrix first order differential operators linearly depending on a spectral parameter. These are AKNS (for Ablowitz, Kaupp, Newell and Segur) with 2 x 2 matrix first order operators; they were generalized by Dubrovin to n x n matrices; we call the latter AKNS-D 50 L.

Automorphic Pseudodifferential Operators 31 Proof. The proof, either by direct calculation, via Jacobi-like forms, or using the Casimir operator, is exactly the same as before. 0 Now we proceed just as in Section 3. The lifting VI< = Ilk V::', gives an isomorphism from M(r) to wDO(R, r)l<, the inverse map being given explicitly by L gn an ....... {ikh»-oo, n«:oo generalizing (4) of Proposition 2, Section 2. On the other hand, it is clear that the product of two conjugate-automorphic wDO's of weight", is again conjugate-automorphic of the same weight, so by transporting the multiplication of wDO's to M(r) by VI< we get a new ring structure {LI< on M(r).

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