Adelic Fourier-Whittaker coefficients and the Casselman-Shalika formula

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 29).In their paper Metaplectic Forms, D. A. Kazhdan and S. J. Patterson developed a generalization of automorphic forms that are defined on metaplectic groups. These groups are non-trivial covering groups of usual algebraic groups, and the forms defined on them are representations that respect the covering. As in the case for automorphic forms, these representations fall into a principle series, indexed by characters on a torus of the metaplectic group, and there is an associated an L-function. In the final section of their paper, an equivalence is shown in the rank one case between this -function and an Dirichlet series defined using Gauss sums, in order to demonstrate the arithmetic content. In this paper we reexamine this connection in the particular case that was discussed in Metaplectic Forms. By looking through the scope of twisted multiplicativity, a property of L-series, the computation is simplified and more easily generalized.by Sawyer Tabony.S.M

Deformations of characters, metaplectic Whittaker functions, and the Yang-Baxter equation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 121-123).Recently, unexpected connections have been discovered between characters of representations and lattice models in statistical mechanics. The bridge was first formed from Kuperberg's solution to the alternating sign matrix (ASM) conjecture. Kuperberg's proof of this conjecture, which enumerates ASMs, utilized a Yang-Baxter equation for a square ice model from statistical mechanics. In earlier work, Tokuyama and okada gave representation theoretic quantities as generating functions on certain symmetry classes of ASMs or generalizations of them. Brubaker, Bump, and Priedberg used a Yang-Baxter equation to reprove Tokuyama's result and this work seeks to do the same for a generalization of Okada's results in type B. We begin by defining the particular lattice model we study. We then imbue the lattice model with Boltzmann weights suggested by a bijection with a set of symmetric ASMs. These weights define a partition function, whose properties are studied by combinatorial and symmetric function methods over the next few chapters. This course of study culminates in the use of the Yang-Baxter equation for our ice model to prove that the partition function factors into a deformation of the Weyl denominator and a generalized character of a highest weight representation, both in type B. We conjecture that the resulting function is connected to metaplectic spherical Whittaker functions. In the last two chapters, we deal with two rather different approaches to computing Whittaker coefficients of metaplectic forms - one using a factorization of the unipotent radical to perform an integration and the other via Hecke operators on the metaplectic group.by Sawyer James Tabony.Ph.D