  The Semiinfinite Flag Space appeared in the works of B.Feigin and E.Frenkel,
and under different disguises was found by V.Drinfeld and G.Lusztig in the
early 80-s. Another recent discovery (Beilinson-Drinfeld Grassmannian) turned
out to conceal a new incarnation of Semiinfinite Flags. We write down these and
other results scattered in folklore. We define the local semiinfinite flag
space attached to a semisimple group $G$ as the quotient $G((z))/HN((z))$ (an
ind-scheme), where $H$ and $N$ are a Cartan subgroup and the unipotent radical
of a Borel subgroup of $G$. The global semiinfinite flag space attached to a
smooth complete curve $C$ is a union of Quasimaps from $C$ to the flag variety
of $G$. In the present work we use $C=P^1$ to construct the category $PS$ of
certain collections of perverse sheaves on Quasimaps spaces, with factorization
isomorphisms. We construct an exact convolution functor from the category of
perverse sheaves on affine Grassmannian, constant along Iwahori orbits, to the
category $PS$. Conjecturally, this functor should correspond to the restriction
functor from modules over quantum group with divided powers to modules over the
small quantum group.
