## 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.## References

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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