Semi-infinite flags and Zastava spaces

We give an interpretation of the semi-infinite intersection cohomology sheaf associated to 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 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-structure 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.