  The prime spectra of two families of algebras, $S^w$ and $\check{S}^w$, $w\in
W,$ indexed by the Weyl group $W$ of a semisimple finitely dimensional are
studied. The algebras $S^w$ have been introduced by A.~Joseph; they are
$q$-analogues of the algebras of regular functions on $w$-translates of the
open Bruhat cell of a semisimple Lie group $G$ corresponding to the Lie algebra
$\fg$.
  We define a stratification of the spectra into components indexed by pairs
$(y_1,y_2)$ of elements of the Weyl group satisfying $y_1\leq w\leq y_2$. Each
component admits a unique minimal ideal which is explicitly described. We show
the inclusion relation of closures to be that induced by Bruhat order.
