In the paper Quantum flag varieties, equivariant quantum
D
\mathcal {D}
-modules, and localization of Quantum groups, Backelin and Kremnizer defined categories of equivariant quantum
O
q
\mathcal {O}_q
-modules and
D
q
\mathcal {D}_q
-modules on the quantum flag variety of
G
G
. We proved that the Beilinson-Bernstein localization theorem holds at a generic
q
q
. Here we prove that a derived version of this theorem holds at the root of unity case. Namely, the global section functor gives a derived equivalence between categories of
U
q
U_q
-modules and
D
q
\mathcal {D}_q
-modules on the quantum flag variety.
For this we first prove that
D
q
\mathcal {D}_q
is an Azumaya algebra over a dense subset of the cotangent bundle
T
⋆
X
T^\star X
of the classical (char
0
0
) flag variety
X
X
. This way we get a derived equivalence between representations of
U
q
U_q
and certain
O
T
⋆
X
\mathcal {O}_{T^\star X}
-modules.
In the paper Localization for a semi-simple Lie algebra in prime characteristic, by Bezrukavnikov, Mirkovic, and Rumynin, similar results were obtained for a Lie algebra
g
p
\mathfrak {g}_p
in char
p
p
. Hence, representations of
g
p
\mathfrak {g}_p
and of
U
q
U_q
(when
q
q
is a
p
p
’th root of unity) are related via the cotangent bundles
T
⋆
X
T^\star X
in char
0
0
and in char
p
p
, respectively.