Explicit doubling integrals for using ``good test vectors''

Author:
Christian A. Zorn

Journal:
Represent. Theory **14** (2010), 285-323

MSC (2010):
Primary 22E50; Secondary 11F70

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

Published electronically:
March 15, 2010

MathSciNet review:
2608965

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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 and where 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 . 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 -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

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

Received by editor(s):
January 9, 2009

Received by editor(s) in revised form:
December 7, 2009

Published electronically:
March 15, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.