Computing Local L-factors for the Unramified Principal Series of Sp(2,F) and its Metaplectic Cover

Abstract

One of the central goals of this thesis is to verify the local Langlands correspondence for the rank two symplectic group Sp(2,F), where F is a p-adic local field. This correspondence seeks to parameterize admissible representations of various matrix groups over F with representations of the Weil-Deligne group of F. This correspondence should include an equality of certain local factors, one being the local L-factors attached to both representations of both the matrix group and the Weil group.

We will restrict our attention to constituents of the unramified principal series of Sp(2,F). In particular, we employ some criteria of Lusztig to assign these representations Weil-Deligne data. While computing the L-factor for representations of the Weil-Deligne group is well known and understood, we require a method for defining the local L-factor for representations of the matrix group.

Our method for defining L-factors for representations of Sp(2,F) is a modification of the doubling integral of Piatetski-Shapiro and Rallis. While Piatetski-Shapiro and Rallis formulate a definition of L-factor via this doubling method, we seek to realize the Weil-Deligne L-factor as an application of our modified integral evaluated on certain ``good test vectors''. Such choices will rely on a wide range of machinery, including intertwining operators, the Weil representation and studying local densities of quadratic form. We tie this wide range of material together, in great detail, through the course of the thesis.

Finally, this method of defining L-factors can be extended in a natural way to representations of the metaplectic cover of Sp(2,F). While the Local Langlands correspondence does not apply to this group, we are still able to produce Weil-Deligne data and L-factors for these representations by using Lusztig's criteria on constituents of the unramified principal series of SO(5,F). In particular, we demonstrate a bijection between constituents of the genuine unramified principal series of Mp(2,F) and the unramified principal series of SO(5,F) in such a way that the doubling L-factor for a representation on the metaplectic group matches the Weil-Deligne L-factor for the corresponding representation on the special orthogonal group.