We give an interpretation of the semi-infinite intersection cohomology sheaf associated with a semisimple simply connected algebraic group in terms of finite-dimensional geometry. Specifically, we describe a procedure for building factorization spaces over moduli spaces of finite subsets of a curve from factorization spaces over moduli spaces of divisors, and show that, under this procedure, the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology (IC) sheaf in the factorizable Grassmannian. We define ‘semi-infinite t-structures’ for a large class of schemes with an action of the multiplicative group, and show that, for the Zastava, the limit of these t-structures recovers the infinite-dimensional version. As an application, we also construct factorizable parabolic semi-infinite IC sheaves and a generalization (of the principal case) to Kac–Moody algebras.
