Hall algebras of cyclic quivers and q-deformed Fock spaces

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel-Hall algebra D(Q) of a quiver Q as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation L(Λ0) of D(Δn) of the cyclic quiver Δn provides a realization of the q-deformed Fock space ⋀∞ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of Δn, we obtain a decomposition of the basic representation L(Λ0) which induces the Kashiwara-Miwa-Stern decomposition of ⋀∞ and a construction of the canonical basis of ⋀∞ defined by Leclerc and Thibon in terms of certain monomial basis elements in D(Δn).