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**Example text**

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.