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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**W. Casselman.*Introduction to the theory of admissible representations of p-adic reductive groups*. Preprint. Accessed at http://www.math.ubc.ca/~cass/research.html.**2.**Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen Rallis,*Explicit constructions of automorphic 𝐿-functions*, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR**892097****3.**Stephen S. Kudla,*Seesaw dual reductive pairs*, Automorphic forms of several variables (Katata, 1983) Progr. Math., vol. 46, Birkhäuser Boston, Boston, MA, 1984, pp. 244–268. MR**763017****4.**S. Kudla. On the Theta Correspondence. Lectures at European School of Group Theory, Beilngries 1996. Accessed at http://www.math.toronto.edu/~skudla/ssk.research.html.**5.**Stephen S. Kudla, Michael Rapoport, and Tonghai Yang,*Modular forms and special cycles on Shimura curves*, Annals of Mathematics Studies, vol. 161, Princeton University Press, Princeton, NJ, 2006. MR**2220359****6.**S. Rallis,*On the Howe duality conjecture*, Compositio Math.**51**(1984), no. 3, 333–399. MR**743016****7.**R. Ranga Rao,*On some explicit formulas in the theory of Weil representation*, Pacific J. Math.**157**(1993), no. 2, 335–371. MR**1197062****8.**J.-P. Serre,*A course in arithmetic*, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR**0344216****9.**T. A. Springer,*Reductive groups*, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR**546587****10.**Marko Tadić,*Jacquet modules and induced representations*, Math. Commun.**3**(1998), no. 1, 1–17 (English, with English and Croatian summaries). MR**1648862****11.**Tonghai Yang,*An explicit formula for local densities of quadratic forms*, J. Number Theory**72**(1998), no. 2, 309–356. MR**1651696**, https://doi.org/10.1006/jnth.1998.2258**12.**Tonghai Yang,*Local densities of 2-adic quadratic forms*, J. Number Theory**108**(2004), no. 2, 287–345. MR**2098640**, https://doi.org/10.1016/j.jnt.2004.05.002**13.**C. Zorn. Computing local -factors for the unramified principal series of and its metaplectic cover. Univ. of Maryland Thesis, 2007.**14.**C. Zorn. Reducibility of the principal series for over a -adic field. Canadian Journal of Mathematics, to appear. Available online at http://www.math.ohio-state.edu/~czorn/works.html.**15.**C. Zorn. Explicit computations of the doubling integral for and . Preprint. Available online at http://www.math.ohio-state.edu/~czorn/works.html.

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2010):
22E50,
11F70

Retrieve articles in all journals with MSC (2010): 22E50, 11F70

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.