New series in the Johnson cokernels of the mapping class groups of surfaces

Let Sigma(g, 1) be a compact oriented surface of genus g with one boundary component, and M-g,M- 1 its mapping class group. Morita showed that the image of the kth Johnson homomorphism tau(M)(k) M-g,M- 1 is contained in the kernel h(g, 1) (k) of an Sp-equivariant surjective homomorphism H circle times(Z) L-2 g (k+1) -> L-2g (k+2), where H := H-1 (Sigma(g, 1), Z) and L-2g (k) is the degree k part of the free Lie algebra L-2g generated by H. In this paper, we study the Sp-module structure of the cokernel h(g,1)(Q) (k) /Im (tau(M)(k,Q)) of the rational Johnson homomorphism tau(M)(k,Q) := tau(M)(k) circle times id(Q), where h(g,1)(Q) (k):= h(g,1) (k) circle times(Z)Q . In particular, we show that the irreducible Sp- module corresponding to a partition [1(k)] appears in the kth Johnson cokernel for any k 1 (mod 4) and k >= 5 with multiplicity one. We also give a new proof of the fact due to Morita that the irreducible Sp- module corresponding to a partition O k _ appears in the Johnson cokernel with multiplicity one for odd k >= 3. The strategy of the paper is to give explicit descriptions of maximal vectors with highest weight [1(k)] and [k] in the Johnson cokernel. Our construction is inspired by the Brauer-Schur-Weyl duality between Sp(2g, Q) and the Brauer algebras, and our previous work for the Johnson cokernel of the automorphism group of a free group.