This dissertation consists of several projects. In the first we introduce the notions of Demazure descent data (DDD) for a triangulated category. This is a collection of functors satisfying a categorical version of degenerate Hecke algebra relations. For such data we define the descent category which can be seen as a categorical version of taking invariants. We construct such data explicitly for the derived category of representations of a Borel subgroup B of an reductive algebraic group G and more generally for the derived category of B-equivariant quasi-coherent sheaves on a G-scheme X. We prove that the derived categories of representations of G and of G-equivariant quasi-coherent sheaves on X is equivalent to their respective descent categories. The result for quasi-coherent sheaves is a categorification of a result in K-theory by Harada, Landweber and Sjamaar.
The next project studies the absolute derived category of equivariant matrix factorizations, where the potential is induced by the moment map of the Hamiltonian action of G on the cotangent bundle of a smooth complex G-variety. Combining results of Isik and Polishchuk-Vaintrob one obtains that in the non-equivariant setting this category is equivalent to the derived category of coherent sheaves on the zero scheme of the moment map. We prove that this result extends to the equivariant setting. This provides an equivalence between the equivariant absolute derived category of matrix factorizations and the derived category of coherent sheaves on the Hamiltonian reduction.
The last project is inspired a by a construction by Bezrukavnikov and Riche of a categorical action of the affine braid group on the equivariant derived category of coherent sheaves on both the Grothendieck and the Springer variety. We use this result to construct such an action on the equivariant absolute derived category of a slightly modified version of the matrix factorizations studied in the previous project.