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Eisenstein series on covers of odd orthogonal groups
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 137, Number 4, August 2015
- pp. 953-1011
- 10.1353/ajm.2015.0031
- Article
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We study the Whittaker coefficients of the minimal parabolic Eisenstein series on the $n$-fold cover of the split odd
orthogonal group ${\rm SO}_{2r+1}$. 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 element 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.