  $G$ be a connected reductive group over an algebraically closed field $k$,
set $K:=k((t))$, let $\gamma\in G(K)$ be a regular semisimple element, let $Fl$
be the affine flag variety of $G^{sc}$, and let $Fl_{\gamma}\subset Fl$ be the
affine Springer fiber at $\gamma$. For every element $w$ of the affine Weyl
group $\widetilde{W}$ of $G$, we denote by $Fl^{\leq w}\subset Fl$ the
corresponding affine Schubert variety, and set $Fl_{\gamma}^{\leq
w}:=Fl_{\gamma}\cap Fl^{\leq w}\subset Fl_{\gamma}$.
  The main result of this paper asserts that if
$w_1,\ldots,w_n\in\widetilde{W}$ are sufficiently regular, then the natural map
$H_i(\cup_{j=1}^n Fl^{\leq w_j}_{\gamma})\to H_i(Fl_{\gamma})$ is injective for
every $i\in\mathbb Z$. This result plays an important role in our work [BV]. To
prove the result, we show that every affine Schubert variety can be written as
an intersection of closures of $U_B(K)$-orbits, where $B$ runs over Borel
subgroups containing a fixed maximal torus $T$, and $U_B$ denotes the unipotent
radical of $B$.
