  We study quantizations of transverse slices to Schubert varieties in the
affine Grassmannian. The quantization is constructed using quantum groups
called shifted Yangians --- these are subalgebras of the Yangian we introduce
which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type.
Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a
quotient of the shifted Yangian quantizes a scheme supported on the transverse
slices, and we formulate a conjectural description of the defining ideal of
these slices which implies that the scheme is reduced. This conjecture also
implies the conjectural quantization of the Zastava spaces for PGL(n) of
Finkelberg-Rybnykov.
