The Hall algebra of a cyclic quiver and canonical bases of the Fock space

where x ∈ Ŝk is minimal such that ν = λ.x −1 satisfies νi < νi+1 for i = 1, 2 . . . k−1 and νi−νk ≥ 1−k−n, and μ = λ.xy. This conjecture is proved by Kazhdan-Lusztig [KL] and Kashiwara-Tanisaki [KT]. The proof relies on an equivalence between the category of finite-dimensional Uǫ(slk)-modules and a category of negative-level representations of the affine algebra ŝlk which are integrable with respect to slk. In [VV], Varagnolo and Vasserot propose a new approach to this conjecture, based on the geometric constructions of simple finite-dimensional Uq(ŝlk)-modules of [GV] and the theory of canonical bases. Let U − n be the generic Hall algebra of the cyclic quiver of type A (1) n−1 (defined over the ring C[q, q ]) and let B be the intersection cohomology basis of Un . Let Λ ∞ be the Fock space representation of Un (see [KMS] and [VV]), and let B be the Leclerc-Thibon canonical bases of Λ (see [LT]). Varagnolo and Vasserot show that the Lusztig formula follows from the equality B|0〉 = B, where |0〉 is the vacuum vector of Λ. In this paper, we give a direct proof of this equality, which can be thought of as a q-analogue of the Lusztig conjecture. This also yields a proof of the positivity conjecture for the basis B (see [LLT1], Conjecture 6.9 i)). Note that the equality B|0〉 = B does not follow from the general theory developped in [K] or