The Hall algebra of a cyclic quiver and canonical bases of Fock spaces

We show that the Ringel-Hall algebra $U_n^-$ of the cyclic quiver of type $A_{n-1}^{(1)}$ is equal to the tensor product of the (negative nilpotent) quantized enveloping algebra $U_q^-{({\widehat{\text{s}\ell}}_n)}$ with a polynomial algebra $C{\lbrack q,q^{-1}\rbrack}{\lbrack z_1,z_2,\dots\rbrack}$. We use this to prove the conjecture of Varagnolo and Vasserot relating the “geometric” canonical basis of the level-1 Fock space (defined via the action of the Hall algebra) with the Leclerc-Thibon basis. In particular, this completes the approach by Varagnolo and Vasserot to a q-analogue of the Lusztig character formula for simple modules over quantum $\mathcal{g}\ell{(k)}$ at roots of unity. Finally, we generalize the previous results to the higher level Fock spaces defined by Uglov.