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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Automorphic Pseudodifferential Operators and an opemtor C-k : R £-k (f) - -+ 21 DO(R)k by ~ (2k - n)! f(n) ak- n ~ n! (k - n)! (k - n - I)! ' and set £0(1) = f. 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)!

13) when a = b = c, but so far we could not find a formula having either one of these properties. 6. Residues, Duality, and Symmetry In the previous section we found a striking symmetry among the three weights k, l, and m := 1 - k - l - n in the formulas giving the coefficients of the nth bracket [J, g]n (f E M2k , 9 E M 2t) in the various multiplications on M(f). To explain it, we use the non-commutative residue map m where Ol(R) = Rdz denotes the space of formal differentials J(z) dz (f E R) and dOO(R) = dR the subspace of exact differentials J'(z)dz, J E R.

8w- 1 . 5) is multiplication by w(w + 1), so if there is any equivariant splitting of this sequence then the lift 1/J of I E R to wDO(R)w must be an eigenvector of C with eigenvalue w(w + 1). Writing w = -k and 1/J(z) = ~:=oJn(Z)8-k-n, we find [C - k(k - I)J 1/J = L [n(n + 2k - 1) In 00 n=l +(n + k)(n + k - 1) J~-l] 8- k- n , 24 Paula Beazley Cohen, Yuri Manin, and Don Zagier and equating all coefficients of this to 0 we find by induction that each In is a multiple of the nth derivative I(n) with coefficients as given in Proposition 1.

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