Semi-infinite flags and Zastava spaces

The semi-infinite flag variety of Feigin and Frenkel is an infinite-dimensional generalization
of the ordinary flag variety associated to a reductive group. Although its sheaf theory is
expected to have deep connections with representations of Kac-Moody algebras and quantum
groups, the infinite-dimensional nature of the space has made constructing its category of D-
modules a difficult task. I will review an early construction of the category using spaces of
based maps from the projective line into the ordinary flag variety (aka ”Zastava spaces”), as
well as a recent construction by Gaitsgory via the semi-infinite intersection cohomology sheaf
on the affine Grassmannian. Following that, I will explain how to modify the geometry of
the Zastava space so that its ordinary intersection cohomology sheaf becomes equivalent, in a
sense which I will make precise, to the semi-infinite IC sheaf.