Eisenstein Series on Covers of Odd Orthogonal Groups

TL;DR

This paper proves a conjecture relating Whittaker coefficients of Eisenstein series on covers of odd orthogonal groups to crystal graph theory, using automorphic and combinatorial methods, and extends results to even degree covers.

Contribution

It establishes the conjecture connecting Whittaker coefficients to crystal graphs for odd degree covers and provides a formula for even degree cases, confirming a broader conjecture.

Findings

Proved the conjecture for odd degree covers using automorphic and combinatorial methods.

Derived a formula for Whittaker coefficients in even degree covers.

Confirmed a Lie-theoretic description of coefficients for large n.

Abstract

We study the Whittaker coefficients of the minimal parabolic Eisenstein series on the $n$-fold cover of the split odd orthogonal group $SO_{2r+1}$. If the degree of the cover is odd, then Beineke, Brubaker and Frechette have conjectured that the $p$-power contributions to the Whittaker coefficients may be computed using the theory of crystal graphs of type C, by attaching to each path component a Gauss sum or a degenerate Gauss sum depending on the fine structure of the path. We establish their conjecture using a combination of automorphic and combinatorial-representation-theoretic methods. Surprisingly, we must make use of the type A theory, and the two different crystal graph descriptions of Brubaker, Bump and Friedberg available for type A based on different factorizations of the long word into simple reflections. We also establish a formula for the Whittaker coefficients in the even degree cover case, again based on crystal graphs of type C. As a further consequence, we establish a Lie-theoretic description of the coefficients for $n$ sufficiently large, thereby confirming a conjecture of Brubaker, Bump and Friedberg.