Explicit doubling integrals for ̃𝑆𝑝₂(ℚ₂) using “good test vectors”

Zorn, Christian

doi:10.1090/S1088-4165-10-00371-7

Explicit doubling integrals for $\widetilde {\mathrm {Sp}_2}(\mathbb {Q}_2)$ using “good test vectors”
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by Christian A. Zorn

Represent. Theory 14 (2010), 285-323

DOI: https://doi.org/10.1090/S1088-4165-10-00371-7

Published electronically: March 15, 2010

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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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