Applications of noncommutative algebraic geometry to representation theory

In this dissertation, we recall some basic notions and results in non-commutative algebraic geometry, especially in spectral theory of abelian categories as developed in [49], [50], [55], [56] and [57]. We apply them to study D-modules on the flag variety X = G/B and its quantized analogue. Via a locality theorem, we reduce the study of D-modules on the flag variety X to the study of modules over Weyl algebras. Then we present three explicit constructions of different classes of irreducible modules over Weyl algebras. For the quantized enveloping algebra Uq( g ), we recover the construction of highest weight irreducible modules, via the Harish-Chandra homomorphism, in the framework of spectral theory. The rest of this work is based on a quantum analogue Xq of the flag variety X and the Beilinson-Bernstein localization for quantized enveloping algebras Uq( g ) as constructed in [41]. The quantum analogue of the Beilinson-Bernstein localization for Uq( g ) reduces the study of representations of Uq( g ) to the study of quantum D-modules on the quantum flag variety Xq. So a natural question is to understand quantum D-modules on Xq. We study quantum D-modules on the quantum flag variety Xq via spectral theory. We show that the locality theorem also applies to the case of quantum D-modules, hence the study of quantum D-modules can be reduced to the study of modules over certain algebras (which are analogues of Weyl algebras, and correspond to Weyl algebras under the classical limit as q → 0). Finally, we study generalized Harish-Chandra objects which is a version of holonomic D-modules in non-commutative algebraic geometry. They coincide by Roos' theorem ([7]) with holonomic D-modules in the case of commutative smooth varieties, in particular, on the classical flag varieties. This notion will be applied to study quantum D-modules on the quantum flag variety Xq.