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THE DOUBLE COVER OF ODD GENERAL SPIN GROUPS, SMALL REPRESENTATIONS, AND APPLICATIONS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  07 August 2015

Eyal Kaplan
Eyal Kaplan*
Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot 7632700, Israel ([email protected])
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Abstract

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We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of $\mathit{SO}_{2n+1}$ of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

Keywords

small representationsmetaplectic coverGSpin groupsco-period integral

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F27: Theta series; Weil representation; theta correspondences
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 3 , June 2017 , pp. 609 - 671
Copyright
© Cambridge University Press 2015 

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