UMD Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/3

New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.

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    The Twining Character Formula for Split Groups and a Cellular Paving for Quasi-split Groups
    (2024) Hopper, Jackson; Haines, Thomas; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The 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.
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    Frame Multiplication Theory for Vector-valued Harmonic Analysis
    (2014) Andrews, Travis David; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    A tight frame is a sequence in a separable Hilbert space satisfying the frame inequality with equal upper and lower bounds and possessing a simple reconstruction formula. We define and study the theory of frame multiplication in finite dimensions. A frame multiplication for a frame is a binary operation on the frame elements that extends to a bilinear vector product on the entire Hilbert space. This is made possible, in part, by the reconstruction property of frames. The motivation for this work is the desire to define meaningful vector-valued versions of the discrete Fourier transform and the discrete ambiguity function. We make these definitions and prove several familiar harmonic analysis results in this context. These definitions beget the questions we answer through developing frame multiplication theory. For certain binary operations, those with the Latin square property, we give a characterization of the frames, in terms of their Gramians, that have these frame multiplications. Combining finite dimensional representation theory and Naimark's theorem, we show frames possessing a group frame multiplication are the projections of an orthonormal basis onto the isotypic components of the regular representations. In particular, for a finite group G, we prove there are only finitely many inequivalent frames possessing the group operation of G as a frame multiplication, and we give an explicit formula for the dimensions in which these frames exist. Finally, we connect our theory to a recently studied class of frames; we prove that frames possessing a group frame multiplication are the central G-frames, a class of frames generated by groups of operators on a Hilbert space.