Algebraic Aspects of Integrable Systems: In Memory of Irene by Mark S. Alber, Gregory G. Luther (auth.), A. S. Fokas, I. M.

By Mark S. Alber, Gregory G. Luther (auth.), A. S. Fokas, I. M. Gelfand (eds.)

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In the supercase, the group r acts on SIItDO(R) via its action on Rand on D. 1) transforms D into (cz+d+o()D. The ring SIItDO(R)f denotes the r-invariant elements of SIItDO(R). We have the filtration SIItDO(R)~, k E Z, inherited from the filtration of SIItDO(R). 8) for the supercase is the sequence, which is split short exact by Theorem 3 below, involving supermodular forms of weight k (for all parities of k) o ~ SIItDO(R)~k_l ~ SIItDO(R)~k ~ SMk(r) ~ o. 4) The analogue of Proposition 1 for supermodular forms is as follows.

7. Supermodular Forms and Superpseudodifferential Operators We work on the supercomplex plane ((;111 with local coordinate (z, (), where (2 = 0, and with the canonical supersymmetric (SUSY) structure given by the maximal non-integrable structure distribution of rank 011 generated by the vector field D = + satisfying D2 = This is, up to isomorphism, the unique such SUSY structure extendible to p111. ) If (z, () is another local coordinate defining the same SUSY structure, then D = J. jj where J = D( () is the superanalogue of the usual Jacobian.

5). The considerations of Section 3-6 about the multiplications of modular forms induced by the multiplications of various kinds of automorphic llfDO's can be generalized in the more or less obvious way (thus an automorphic super-llfDO of mixed weight is just a super-llfDO which is multiplied on the left and on the right by some powers of J (A) under the action of A E r), and the arguments given in the last section can also be generalized using the superversion of the non-commutative residue map given in [MRJ.

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