Canonical Bases for Quantum Groups and Whittaker Coefficients of Metaplectic Eisenstein Series

Abstract

This thesis explores the connection between Whittaker coefficients of metaplectic Eisenstein series on algebraic groups and the canonical basis for the quantized universal enveloping algebra of the associated Langlands dual group. Whittaker coefficients of Eisenstein series are often described in terms of the representation theory of the dual group; this be observed in the classic Casselman-Shalika formula for the Whittaker coefficients of the non-metaplectic Borel Eisenstein series in terms of characters of highest weight representations of the dual group. When we pass to Eisenstein series on metaplectic covers of algebraic groups, often more intricate representation theoretic information is necessary. For example, in [1] Brubaker and Friedberg calculate Whittaker coefficients of maximal parabolic Eisenstein series in the metaplectic setting. They make use of a parameterization of coset representatives of the big cell of a Bruhat decomposition, and connect this to Lusztig’s parameterization of the canonical basis. We will state a conjecture that certain formulas coming from the coset parameterization are highest weight inequalities on Lusztig data for nice decompositions, generalizing a conjecture of Brubaker and Friedberg from [1]. We prove this conjecture in type A, and provide evidence for the conjecture in other types.