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Representation Theory

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ISSN 1088-4165

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Explicit doubling integrals for $\widetilde {\mathrm {Sp}_2}(\mathbb {Q}_2)$ using “good test vectors”
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by Christian A. Zorn PDF
Represent. Theory 14 (2010), 285-323 Request permission

Abstract:

In a previous paper (see http:/www.math.ohio-state.edu/~czorn/works.html), we computed examples of the doubling integral for constituents of the unramified principal series of $\mathrm {Sp}_2(F)$ and $\widetilde {\textrm {Sp}_2}(F)$ where $F$ was a non-dyadic field. These computations relied on certain “good test vectors” and “good theta test sections” motivated by the non-vanishing of theta lifts. In this paper, we aim to prove a partial analog for $\widetilde {\textrm {Sp}_2}(\mathbb {Q}_2)$. However, due to several complexities, we compute the doubling integral only for certain irreducible principal series representations induced from characters with ramified quadratic twists. We develop some $2$-adic analogs for the machinery in the paper mentioned above; however, these tend to be more delicate and have more restrictive hypotheses than the non-dyadic case. Ultimately, this paper and the one mentioned above develop several computations intended to be used for future research into the non-vanishing of theta lifts.
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Additional Information
  • Christian A. Zorn
  • Affiliation: Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210
  • Email: czorn@math.ohio-state.edu
  • Received by editor(s): January 9, 2009
  • Received by editor(s) in revised form: December 7, 2009
  • Published electronically: March 15, 2010
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 14 (2010), 285-323
  • MSC (2010): Primary 22E50; Secondary 11F70
  • DOI: https://doi.org/10.1090/S1088-4165-10-00371-7
  • MathSciNet review: 2608965