Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The nonarchimedean theta correspondence
for $\operatorname{GS\lowercase{p}}(2)$ and $\operatorname{GO}(4)$

Author: Brooks Roberts
Journal: Trans. Amer. Math. Soc. 351 (1999), 781-811
MSC (1991): Primary 11F27; Secondary 22E50
MathSciNet review: 1458334
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the theta correspondence between the sets $\operatorname{Irr} (\operatorname{GSp} (2,k))$ and $\operatorname{Irr} (\operatorname{GO} (X))$ when $k$ is a nonarchimedean local field and $\dim _{k} X =4$. Our main theorem determines all the elements of $\operatorname{Irr} (\operatorname{GO} (X))$ that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of $\operatorname{Irr} (\operatorname{O} (X))$ that occur in the theta correspondence between $\operatorname{Irr} (\operatorname{Sp} (2,k))$ and $\operatorname{Irr} (\operatorname{O} (X))$. We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.

References [Enhancements On Off] (What's this?)

  • [BZ] I.N. Bernshtein and A.V. Zelevinskii, Representations of the group $\operatorname {Gl}(n, F)$ where $F$ is a nonarchimedean local field, Russian Math. Surveys 31 (1976), 1-68. MR 54:12988
  • [C] P. Cartier, Representations of $\mathfrak{p}$-adic groups: a survey, in Automorphic Forms, Representations, L-functions, Proc. Symposia Pure Math. vol. XXXIII - Part 1, American Mathematical Society, Providence, 1979. MR 81e:22029
  • [Ca] W. Casselman, On the representations of $Sl_{2} (k)$ related to binary quadratic forms, Amer. J. Math. 94 (1972), 810-834. MR 47:6948
  • [Co] M. Cognet, Representation de Weil et changement de base quadratique, Bull. Soc. Math. France 113 (1985), 403-457. MR 88h:22029a
  • [F] Y. Flicker, On distinguished representations, J. Reine Angew. Math. 418 (1991), 139-172. MR 92i:22019
  • [G] S.S. Gelbart, Automorphic Forms on Adele Groups, Annals of Mathematics Studies 83, Princeton University Press, Princeton, 1975. MR 52:280
  • [GK] S.S. Gelbart and A.W. Knapp, L-indistinguishability and $R$ groups for the special linear group, Adv. in Math. 43 (1982), 101-121. MR 83j:22009
  • [GL] P. Géradin and J.P. Labesse, The solution of a base change problem for $\operatorname {Gl}(2)$ (following Langlands, Saito, Shintani), in Automorphic Forms, Representations, L-functions, Proc. Symposia Pure Math. vol. XXXIII - Part 2, American Mathematical Society, Providence, 1979. MR 82e:10047
  • [GRS] S. Gelbart, J. Rogawski and D. Soudry, On periods and algebraic cycles for $\operatorname {U}(3)$, Israel J. of Math. 83 (1993), 213-252. MR 95a:11047
  • [H] J. Hakim, Distinguished p-adic representations, Duke Math. J. 66 (1991), 1-22. MR 92c:22037
  • [HK] M. Harris and S.S. Kudla, The central critical value of a triple product L-function, Annals of Math. 133 (1991), 605-672. MR 93a:11043
  • [HKS] M. Harris, S.S. Kudla and W.J. Sweet, Theta dichotomy for unitary groups, Preprint (1994). MR 96m:11041
  • [HLR] G. Harder, R. Langlands and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flachen, J. Reine Angew. Math. 366 (1986), 53-120. MR 87k:11066
  • [HPS] R. Howe and I.I. Piatetski-Shapiro, Some examples of automorphic forms on $\operatorname {Sp}_{4}$, Duke Math. J. 50 (1983), 55-106. MR 84m:10019
  • [HST] M. Harris, D. Soudry and R. Taylor, $l$-adic representations associated to modular forms over imaginary quadratic fields I: lifting to $\operatorname {GSp}_{4}(\mathbb{Q})$, Invent. Math. 112 (1993), 377-411. MR 94d:11035
  • [K] S.S. Kudla, On the local theta correspondence, Invent. Math. 83 (1986), 229-255. MR 87e:22037
  • [KR] S.S. Kudla and S. Rallis, A regularized Siegel-Weil formula: the first term identity, Annals of Math. 140 (1994), 1-80. MR 95f:11036
  • [MVW] C. Moeglin, M.-F. Vigneras and J.-L. Waldspurger, Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Mathematics 1291, Springer-Verlag, Berlin-Heidelberg-New York, 1987. MR 91f:11040
  • [O] O.T. O'Meara, Introduction to Quadratic Forms, Springer-Verlag, Berlin-Heidelberg-New York, 1973. MR 50:269
  • [P1] D. Prasad, Trilinear forms for representations of $\operatorname {Gl}(2)$ and local $\epsilon $-factors, Compositio Math. 75 (1990), 1-46. MR 91i:22023
  • [P2] D.Prasad, Invariant forms for representations of $\operatorname {Gl}_{2}$ over a local field, Amer. J. Math. 114 (1992), 1317-1363. MR 93m:22011
  • [PSS] I.I. Piatetski-Shapiro and D. Soudry, $L$ and $\varepsilon $ factors for $\operatorname {GSp}(4)$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1982), 505-530. MR 84m:22029
  • [R] B. Roberts, The theta correspondence for similitudes, Israel J. of Math. 94 (1996), 285-317. MR 98a:22007
  • [Ra] S. Rallis, On the Howe duality conjecture, Compositio Math. 51 (1984), 333-399. MR 85g:22034
  • [Sh] D. Shelstad, Notes on $L$-indistinguishability, in Automorphic Forms, Representations, L-functions, Proc. Symposia Pure Math. vol. XXXIII - Part 2, American Mathematical Society, Providence, 1979. MR 81m:12018
  • [S] H. Shimizu, Theta series and automorphic forms on $\operatorname {Gl}(2)$, J. Math. Soc. Japan 24 (1972), 638-683. MR 54:12658
  • [Sc] W. Scharlau, Quadratic and Hermitian Forms, Springer-Verlag, Berlin-Heidelberg-New York, 1985. MR 86k:11022
  • [So1] D. Soudry, The $L$ and $\gamma $ factors for generic representations of $\operatorname {GSp}(4,k) \times \operatorname {Gl}(2,k)$ over a local nonarchimedean field $k$, Duke Math. J. 51 (1984), 355-394. MR 86f:22022
  • [So2] D. Soudry, A uniqueness theorem for representations of $\operatorname {GSO}(6)$ and the strong multiplicity one theorem for generic representations of $\operatorname {GSp}(4)$, Israel J. of Math. 58 (1987), 257-287. MR 89f:22025
  • [T] J. Tunnell, Local $\epsilon $-factors and characters of $\operatorname {Gl}(2)$, Amer. J. Math. 105 (1983), 1277-1307. MR 86a:22018
  • [V] M.F. Vignéras, Correspondances entre representations automorphes de $\operatorname {Gl}(2)$ sur une estension quadratic de $\operatorname {GSp}(4)$ sur $\mathbb{Q}$, conjecture locale de Langlands pour $\operatorname {GSp}(4)$, in The Selberg Trace Formula and Related Topics, American Mathematical Society, Providence, R.I., 1986. MR 88e:11118
  • [W] J.-L. Waldspurger, Demonstration d'une conjecture de duality de Howe dans le case $p$-adiques, $p \neq 2$, in Israel Math. Conf. Proc. vol. 2, 1990, pp. 267-324. MR 93h:22035

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11F27, 22E50

Retrieve articles in all journals with MSC (1991): 11F27, 22E50

Additional Information

Brooks Roberts
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
Address at time of publication: Department of Mathematics, Brink Hall, University of Idaho, Moscow, Idaho 83844-1103

Keywords: Nonarchimedean theta correspondence, \protect{$\GSp (2)$, $\GO (4)$}
Received by editor(s): June 3, 1996
Received by editor(s) in revised form: February 6, 1997
Additional Notes: During the period of this work the author was a Research Associate with the NSF 1992–1994 special project Theta Functions, Dual Pairs, and Automorphic Forms at the University of Maryland, College Park, and was supported by a Stipendium at the Max-Planck-Institut für Mathematik.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society