Hopper, JacksonThe dissertation contains two main results. The first is on the twisted Weyl character formula for split groups and the second is a cellular paving result for convolution morphisms in partial affine flag varieties of quasi-split groups. Let G^ be a connected reductive group over an algebraically closed field of characteristic 0 with a pinning-preserving outer automorphism σ. Jantzen’s twining character formula relates the trace of the action of σ on a highest-weight representation of G^ to the character of a corresponding highest-weight representation of a related group. This paper extends the methods of Hong’s geometric proof for the case G^ is adjoint, to prove that the formula holds for all split, connected, reductive groups, and examines the role of additional hypotheses. In particular, it is shown that for a disconnected reductive group G, the affine Grassmannian of G is isomorphic to the affine Grassmannian of its neutral component. In the final section, it is explained how these results can be used to draw conclusions about quasi-split groups over a non-Archimedean local field. This paper thus provides a geometric proof of a generalization of the Jantzen twining character formula, and provides some apparently new results of independent interest along the way. Now we turn to the context of Chapter 3. Let G be a tamely ramified, quasi-split group over a Laurent series field K = k((t)), where k is either finite or algebraically closed. If k is finite of order q and the split adjoint form of G contains a factor of type D4, then we also assume either 3 divides q or 3 divides q-1. Given a sequence of Schubert varieties contained in a fixed partial affine flag variety F for G, consider the convolution morphism m that maps the twisted product of those Schubert varieties into the partial affine flag variety F. We show that the fibers of m are paved by finite products of affine spaces and punctured affine spaces. This generalizes a result of Haines, which proves a similar result in the case G is split and defined over k. A consequence for structure constants of parahoric Hecke algebras is deduced.enDissertationMathematicsAlgebraic GeometryBruhat-Tits TheoryGeometric Representation TheoryReductive GroupsRepresentation Theory