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Additional resources for Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman
Interpretations and Extensions of the Basic Lifting In this section we discuss some further aspects of the proposition just proved. In particular we describe the relationship between modular forms, invariant \liDO's, and "Jacobi-like forms" (this was the point of view taken in [ZI]) , give a different and more conceptual proof of Proposition 1 in terms of the Casimir operator for sl(2, q, and describe an extension to generalized \liDO's where one allows non-integral powers of a. Jacobi-like forms.
Then Ck(l12kg) = Ck(l)og for all 9 E PSL(2,q and all k E Z. In particular, if f E M 2k(r) for some subgroup r c PSL(2, q then Ck(l) E \lIDO(R)~k. Proof. Write 9 = e ~). By induction on n we obtain the formula dn (II ()) _ ~ n! (k + n dzn kg Z - ~ r! 9) for any k E Z and any n ;::: 0, where f(r) denotes or f as usual. (m+r+k-l)! (2k+r-l)! (cz + d)2k+m cz+d . r,m~O The proof for k < 0, is similar, and the case k = 0 is of course trivial. 0 2. Interpretations and Extensions of the Basic Lifting In this section we discuss some further aspects of the proposition just proved.
3) follows from the following generalization of the identity given in Section 3, and whose proof again will be postponed to the paper [Z2]: Identity. ) Again this identity reveals surprising "hidden symmetries": the left-hand side is symmetric under interchanging y and z and simultaneously replacing a by -a and has no other evident symmetries, but the identity shows that it is in fact symmetric in all three variables x, y, z and at the same time an even 32 Paula Beazley Cohen, Yuri Manin, and Don Zagier function of a.