We study Translation functors and Wall-Crossing functors on inn-nite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem; two important results were established earlier by W. Soergel in a totally diierent way. Other applications to the category O of Bernstein-Gelfand-G...

We develop a general framework to deal with the unitary representations of quantum groups using the language of C∗-algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.

Dilworth s theorem states a duality relation between minimum chain decompositions of a directed acyclic graph and maximum antichains We generalize the theorem to apply when the chains of the decomposition are required to contain the chains of an initial decomposition We show that duality obtains precisely when an associated undirected graph is perfect We apply this result to a vehicle routing a...

Cohomological induction gives an algebraic method for constructing representations of a real reductive Lie group G from irreducible representations of reductive subgroups. Beilinson-Bernstein localization alternatively gives a geometric method for constructing Harish-Chandra modules for G from certain representations of a Cartan subgroup. The duality theorem of Hecht, Miličić, Schmid and Wolf e...

We consider constrained discounted-cost Markov control processes in Bore1 spaces, with unbounded costs. Conditions are given for the constrained problem to be solvable, and also equivalent to an equality-constrained (EC) 1' inear program. In addition, it is shown that there is no duality gap between EC and its dual program EC*, and that, under additional assumptions, also EC* is solvable, so th...

w x In 8 , K. Morita introduced a useful notion of duality between categories of modules, usually called ‘‘Morita duality.’’ He proved that every duality is given by contravariant hom functors defined by a bimodule which is an injective cogenerator for both categories of modules. On the other hand, the equivalences between categories of comodules over a coalgebra w x were characterized by M. Ta...