Abstract
The prime spectra of two families of algebras,
S
w
and
S
ˇ
w
,
w∈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
≤w≤
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.