Grodzicki, William2017-10-092017-10-092017-06https://hdl.handle.net/11299/190561University of Minnesota Ph.D. dissertation. June 2017. Major: Mathematics. Advisor: Benjamin Brubaker. 1 computer file (PDF); iii, 55 pages.We realize the non-split Bessel model of Novodvorsky and Piatetski-Shapiro as a generalized Gelfand-Graev representation of GSp(4), as defined by Kawanaka. Our primary goal is to calculate the values of Iwahori-fixed vectors of unramified principal series representations in the Bessel model. On the path to achieving this goal, we will first use Mackey theory to realize the Bessel functional as an integral - as a result, we will reestablish the uniqueness and existence of a Bessel model for principal series representations, originally proved by Novodvorsky and Piatetski-Shapiro and by Bump, Friedberg, and Furusawa, respectively. Inspired by the work of Brubaker, Bump, and Friedberg, our method of calculation takes advantage of the connection between the Iwahori-fixed vectors in the Bessel model and a certain linear character of the Hecke algebra of GSp(4). We will also provide a detailed description of the conjectural program connecting characters of the Hecke algebra for a more general reductive group G with multiplicity-free models of principal series representations. In particular, we will focus on the role played by the Springer correspondence in this program. Additionally, using the formulas we develop for the Iwahori-fixed vectors, we provide an explicit alternator expression for the spherical vector in the Bessel model which matches previous results of Bump, Friedberg, and Furusawa.enNumber TheoryRepresentation TheoryThesis or Dissertation