Theta Functions on Covers of Symplectic Groups
Solomon Friedberg, David Ginzburg

TL;DR
This paper investigates automorphic theta representations on covers of symplectic groups, establishing their properties and connections, including explicit computations of local factors and conjectured relations between different theta representations.
Contribution
It introduces new constructions of generic representations from theta representations on covering symplectic groups and computes their local factors explicitly.
Findings
Factorizable Whittaker functions for certain representations
Explicit unramified local factors in terms of Gauss sums
Conditional results for odd r with n ≤ r < 2n
Abstract
We study the automorphic theta representation on the -fold cover of the symplectic group . This representation is obtained from the residues of Eisenstein series on this group. If is odd, , then under a natural hypothesis on the theta representations, we show that may be used to construct a generic representation on the -fold cover of . Moreover, when the Whittaker functions of this representation attached to factorizable data are factorizable, and the unramified local factors may be computed in terms of -th order Gauss sums. If we prove these results, which in that case pertain to the six-fold cover of , unconditionally. We expect that in fact the representation constructed here, , is precisely ; that is, we…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic structures and combinatorial models · Algebraic Geometry and Number Theory
