Let Σ g,1 be a compact oriented surface of genus g with one boundary component, and M g,1 its mapping class group. Morita showed that the image of the k-th Johnson homomorphism τ M k of M g,1 is contained in the kernel h g,1 (k) of an Sp-equivariant surjective homomorphism H ⊗ Z L 2g (k + 1) → L 2g (k + 2), where H := H 1 (Σ 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 cokernelIn particular, we show that the irreducible Sp-module corresponding to a partition [1 k ] appears in the k-th 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 [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.

2861On the derivation algebra of the free Lie algebra and trace maps NAOYA ENOMOTO TAKAO SATOHWe mainly study the derivation algebra of the free Lie algebra and the Chen Lie algebra generated by the abelianization H of a free group, and trace maps. To begin with, we give the irreducible decomposition of the derivation algebra as a GL.n; Q/-module via the Schur-Weyl duality and some tensor product theorems for GL.n; Q/. Using them, we calculate the irreducible decomposition of the images of the Johnson homomorphisms of the automorphism group of a free group and a free metabelian group.Next, we consider some applications of trace maps: Morita's trace map and the trace map for the exterior product of H . First, we determine the abelianization of the derivation algebra of the Chen Lie algebra as a Lie algebra, and show that the abelianization is given by the degree one part and Morita's trace maps. Second, we consider twisted cohomology groups of the automorphism group of a free nilpotent group. In particular, we show that the trace map for the exterior product of H defines a nontrivial twisted second cohomology class of it.

In the papers [EK1], [EK2] and [EK3] with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type B. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type B are described by using the lower global basis of symmetric crystals of V θ (λ). In the present paper, we prove the existence of crystal bases and global bases of V θ (0) for any symmetric quantized Kac-Moody algebra by using a geometry of quivers (with a Dynkin diagram involution). This is analogous to George Lusztig's geometric construction of U − v and its lower global basis. 9

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We determine the composition factors of the polynomial representation of DAHA, conjectured by M. Kasatani in [Kasa, Conjecture 6.4.]. He constructed an increasing sequence of subrepresentations in the polynomial representation of DAHA using the "multi-wheel condition", and conjectured that it is a composition series. On the other hand, DAHA has two degenerate versions called the "degenerate DAHA" and the "rational DAHA". The category O of modules over these three algebras and the category of modules over the v-Schur algebra are closely related. By using this relationship, We reduce the determination of composition factors of polynomial representations of DAHA to the determination of the composition factors of the Weyl module W (1 n ) for the v-Schur algebra. By using the LLT-Ariki type theorem of v-Schur algebra proved by Varagnolo-Vasserot, we determine the composition factors of W (1 n ) by calculating the upper global basis and crystal basis of Fock space of U q ( sl ℓ ).