More general setting
The question actually fits into the more general setting that is described in David Ben-Zvi's answer to this MO question (which is about categorification of the Chern character).
HRR for dg algebras
Shklyarov is the one who developped RR theorem for noncommutatve derived schemes (by this one should understand smooth proper DG algebras): http://arxiv.org/abs/0710.1937.
Shklyarov's result has been improved recently by Petit, Lunts, and also Polishchuk-Vaintrob in the context of matrix factorizations.
Let me explain what the statement is. Let $A$ be a proper and homologically smooth dg algebra $A$ (proper means that $\sum_n dim(H^n(A))<+\infty$, and homologicaly smooth means that $A$ is perfect in $D(A\otimes A^{op})$). Let $M$ be a perfect $A$-module. There is a trace map $ch:Hom_{D^{perf}(A)}(M,M)\to HH_0(A)$ (see e.g. this paper of Caldararu-Willerton for a very nice description of the kind of traces I am speaking about), which you can consider as being the Chern character.
Now for an $A$-module $M$ and an $A^{op}$-module $N$ we can consider the $k$-module $N\otimes_AM$ (all my tensor products are derived). Then for $f:M\to M$ and $g:N\to N$ we can consider $ch(g\otimes f)=str(g\otimes f)\in HH_0(k)=k$. Finally the formula is
$$
ch(g)\cup ch(f)=ch(g\otimes f)
$$
where $\cup:HH_*(A^{op})\otimes HH_*(A)\to HH_*(k)=k$ is the so-called (categorical) Mukai pairing. This is actually more a Lefshetz type formula.
The Todd class is actually hidden in the Mukai pairing (the point is that for associative algebras there is no analogon neither for the Todd class, nor for the usual pairing given by integration).
RR for D-modules
To my knowledge the first one who proved a RR Theorem for D-modules is Laumon (Sur la categorie derivee des D-modules filtres, Algebraic Geometry, M. Raynaud and T. Shioda eds, Lecture Notes in Math. Springer-Verlag 1016 pp. 151–237, 1983). Then Schapira and Scheinders also considered it (Index theorem for elliptic pairs II. Euler class and relative index theorem, Asterisque 224 Soc. Math. France, 1994) and made a very important conjecture which has been proved by Bressler-Nest-Tsygan using methods of deformation quantization ( http://arxiv.org/abs/math/9904121 and http://arxiv.org/abs/math/0002115) developped by Fedosov.
There is also a paper of Engeli and Felder that gives a Lefschetz type formula. Their approach has been later clarified by Ramadoss (he has many paper on this subject that you can find on arXiv).
The subject really moved to deformation quantization. You can learn a lot about all this (with also more details on who one should credit for what) in Section 4, 5 and 6 of this book by Kashiwara and Schapira.
If you have a formal noncommutative deformation $A_\hbar$ of the structure sheaf $\mathcal O_X$ (maybe as a twisted resheaf, or algebroid stack - this is what Kashiwara and Schapira call a DQ algebroid), then you can play the same game as with a smooth proper dg algebras : define a trace with values in Hochschild homology, and state a HRR type Theorem (better, Lefshetz formula) about the compatibility of the cup product on Hochschild homology with the composition of kernels. The reason for that relies on some finiteness and duality properties for cohomologicall complete $A_\hbar$-modules. The main difficulty is then to prove such a result.
To conclude this paragraph, let me observe that (the Rees algebra of) $\mathcal D_X$ can be viewed as a deformation of $\mathcal O_{T^*X}$.
Relation between the question for nc schemes and D-modules
Last but not least, the relation between the derived non-commutative geometry and deformation quantization stuff is adressed in a very recent preprint of Petit: his strategy to prove HRR for DQ algebroids is to use the result for smooth proper DG algebras. Namely, he proves that some derived category of cohomologically complete modules over a DQ-algebroid on a projective variety has a compact generator.
This subject is currently very active.
I apologize for that this answer reduces to a (non-exhaustive) list of references. If you have a more specific question I can tell you where to go in these references in order to (hopefully) find an answer.
What I want to ask is there any other
way to define quantized flag variety?
In the classical case, it is well
known that flag variety can be defined
as $G/B$, say $G$ is general linear group
and B is Borel subgroup. Is there any
analogue for quantum case? Is there a
definition like $G_q$ as "quantum linear
group" and $B_q$ as quantum analogue of
Borel subgroup?
The Borel subgroup of the quantized function algebra $G_q$ is of course certain quotient Hopf algebra $p:\mathcal{O}(G_q)\to \mathcal{O}(B_q)$ which then canonically coacts on $G_q$ from both sides (from the left by $(p\otimes id)\circ\Delta_{G_q}:\mathcal{O}(G_q)\to \mathcal{O}(B_q)\otimes \mathcal{O}(G_q)$. In the case of $SL_q(n)$ you get the lower Borel by quotienting by the Hopf ideal generate by the entries of the matrxi $T$ of the generators standing above the diagonal.
Of course, if you define $Qcoh(G_q)$ as the category ${}_{\mathcal{O}(G_q)}\mathcal{M}$ of left modules over $\mathcal{O}(G_q)$ then the category of relative left-right Hopf modules ${}_{\mathcal{O}(G_q)}\mathcal{M}^{\mathcal{O}(B_q)}$ is precisely the category of quasicoherent sheaves over the quantum flag variety $G_q/B_q$. If you look at several of my earlier articles they together show that in the case of $SL_q(n)$ at least, there is a collection of localizations of the above category of relative Hopf modules which are affine, which gives it a structure of a noncommutative scheme with a canonical atlas whose cardinality is the cardinality of the Weyl group (the generalization to other parabolics is easy). Moreover the forgetful functor from the category of Hopf modules to ${}_{\mathcal{O}(G_q)}\mathcal{M}$ is the inverse image part of the geometric morphism which is a Zariski locally trivial $B_q$-fibration; the local triviality, which is obtained with the help of the quantum Gauss decomposition, boils down to the Schneider's equivalence for the extension of the algebras of localized coinvariants into the corresponding localization of $G_q$. The sketch of the whole program is in
- Localizations for construction of quantum coset spaces, in "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003, math.QA/0301090.
For the Peter-Weyl look at the original works of S. Woronowicz or
- A. Klimyk, K. Schmüdgen, Quantum groups and their representations, Springer 1997.
Best Answer
Yes, there exists the result for quantized enveloping algebra. It is developed by Lunts-Rosenberg and proved by them and Tanisaki.
There are several notions:
In the framework of noncommutative algebraic geometry. quantized flag variety is defined as a noncommutative projective scheme. It is a proj-category of quantized enveloping algebra of Lie algebra g. It is a noncommutative separated scheme with affine covers discovered by A.Joseph.
Lunts and Rosenberg defined differential calculus in noncommutative algebraic geometry. They introduced the noncommutative version of Grothendieck differential operators in differential operator on noncommutative ring and then applied this construction to define quantized D-modules in localization for quantum group. In this paper, they formulated the quantized Beilinson Bernstein localization for quantized enveloping algebra in generic case. They proved the global section functor is exact and conjectured it is indeed the correct quantized version, which means it is an equivalent.
Later, under this framework. Tanisaki proved in his paper The Beilinson-Bernstein correspondence for quantized enveloping algebras that the conjecture of Lunts-Rosenberg is indeed true. Moreover, Tanisaki proved this result in root of unity case,see D-modules on quantized flag manifolds at roots of 1.
More comments: In the paper of Lunts-Rosenberg, they pointed out the localization for the quantum group sl2 was constructed by "hand" by T.J.Hodges.