The nonarchimedean theta correspondence for 𝐺𝑆𝑝(2) and 𝐺𝑂(4)

Roberts, Brooks

doi:10.1090/S0002-9947-99-02126-1

The nonarchimedean theta correspondence for $\mathrm {GSp}(2)$ and $\mathrm {GO}(4)$
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Trans. Amer. Math. Soc. 351 (1999), 781-811 Request permission

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

I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70 (Russian). MR 0425030
P. Cartier, Representations of $p$-adic groups: a survey, 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. 111–155. MR 546593
William Casselman, On the representations of $\textrm {SL}_{2}(k)$ related to binary quadratic forms, Amer. J. Math. 94 (1972), 810–834. MR 318401, DOI 10.2307/2373760
Michel Cognet, Représentation de Weil et changement de base quadratique, Bull. Soc. Math. France 113 (1985), no. 4, 403–457 (French, with English summary). MR 850776
Yuval Z. Flicker, On distinguished representations, J. Reine Angew. Math. 418 (1991), 139–172. MR 1111204, DOI 10.1515/crll.1991.418.139
Stephen S. Gelbart, Automorphic forms on adèle groups, Annals of Mathematics Studies, No. 83, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR 0379375
S. S. Gelbart and A. W. Knapp, $L$-indistinguishability and $R$ groups for the special linear group, Adv. in Math. 43 (1982), no. 2, 101–121. MR 644669, DOI 10.1016/0001-8708(82)90030-5
P. Gérardin and J.-P. Labesse, The solution of a base change problem for $\textrm {GL}(2)$ (following Langlands, Saito, Shintani), 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. 115–133. MR 546613
S. Gelbart, J. Rogawski, and D. Soudry, On periods of cusp forms and algebraic cycles for $\textrm {U}(3)$, Israel J. Math. 83 (1993), no. 1-2, 213–252. MR 1239723, DOI 10.1007/BF02764643
Jeff Hakim, Distinguished $p$-adic representations, Duke Math. J. 62 (1991), no. 1, 1–22. MR 1104321, DOI 10.1215/S0012-7094-91-06201-0
Michael Harris and Stephen S. Kudla, The central critical value of a triple product $L$-function, Ann. of Math. (2) 133 (1991), no. 3, 605–672. MR 1109355, DOI 10.2307/2944321
Michael Harris, Stephen S. Kudla, and William J. Sweet, Theta dichotomy for unitary groups, J. Amer. Math. Soc. 9 (1996), no. 4, 941–1004. MR 1327161, DOI 10.1090/S0894-0347-96-00198-1
G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math. 366 (1986), 53–120 (German). MR 833013
Roger Howe and I. I. Piatetski-Shapiro, Some examples of automorphic forms on $\textrm {Sp}_{4}$, Duke Math. J. 50 (1983), no. 1, 55–106. MR 700131
Michael Harris, David Soudry, and Richard Taylor, $l$-adic representations associated to modular forms over imaginary quadratic fields. I. Lifting to $\textrm {GSp}_4(\textbf {Q})$, Invent. Math. 112 (1993), no. 2, 377–411. MR 1213108, DOI 10.1007/BF01232440
Stephen S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), no. 2, 229–255. MR 818351, DOI 10.1007/BF01388961
Stephen S. Kudla and Stephen Rallis, A regularized Siegel-Weil formula: the first term identity, Ann. of Math. (2) 140 (1994), no. 1, 1–80. MR 1289491, DOI 10.2307/2118540
Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger, Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). MR 1041060, DOI 10.1007/BFb0082712
O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Band 117, Springer-Verlag, New York-Heidelberg, 1971. Second printing, corrected. MR 0347768
Dipendra Prasad, Trilinear forms for representations of $\textrm {GL}(2)$ and local $\epsilon$-factors, Compositio Math. 75 (1990), no. 1, 1–46. MR 1059954
Dipendra Prasad, Invariant forms for representations of $\textrm {GL}_2$ over a local field, Amer. J. Math. 114 (1992), no. 6, 1317–1363. MR 1198305, DOI 10.2307/2374764
Ilya I. Piatetski-Shapiro and David Soudry, $L$ and $\varepsilon$ factors for $\textrm {GSp}(4)$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 505–530 (1982). MR 656034
Brooks Roberts, The theta correspondence for similitudes, Israel J. Math. 94 (1996), 285–317. MR 1394579, DOI 10.1007/BF02762709
S. Rallis, On the Howe duality conjecture, Compositio Math. 51 (1984), no. 3, 333–399. MR 743016
D. Shelstad, Notes on $L$-indistinguishability (based on a lecture of R. P. Langlands), 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. 193–203. MR 546618
Hideo Shimizu, A correction to: “Theta series and automorphic forms on GL$_{2}$” (J. Math. Soc. Japan 24 (1972), 638–683), J. Math. Soc. Japan 26 (1974), 374–376. MR 424699, DOI 10.2969/jmsj/02620374
Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR 770063, DOI 10.1007/978-3-642-69971-9
David Soudry, The $L$ and $\gamma$ factors for generic representations of $\textrm {GSp}(4,\,k)\times \textrm {GL}(2,\,k)$ over a local non-Archimedean field $k$, Duke Math. J. 51 (1984), no. 2, 355–394. MR 747870, DOI 10.1215/S0012-7094-84-05117-2
David Soudry, A uniqueness theorem for representations of $\textrm {GSO}(6)$ and the strong multiplicity one theorem for generic representations of $\textrm {GSp}(4)$, Israel J. Math. 58 (1987), no. 3, 257–287. MR 917359, DOI 10.1007/BF02771692
Jerrold B. Tunnell, Local $\epsilon$-factors and characters of $\textrm {GL}(2)$, Amer. J. Math. 105 (1983), no. 6, 1277–1307. MR 721997, DOI 10.2307/2374441
Marie-France Vignéras, Correspondances entre representations automorphes de $\textrm {GL}(2)$ sur une extension quadratique de $\textrm {GSp}(4)$ sur $\textbf {Q}$, conjecture locale de Langlands pour $\textrm {GSp}(4)$, The Selberg trace formula and related topics (Brunswick, Maine, 1984) Contemp. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 463–527 (French). MR 853573, DOI 10.1090/conm/053/853573
J.-L. Waldspurger, Démonstration d’une conjecture de dualité de Howe dans le cas $p$-adique, $p\neq 2$, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 267–324 (French). MR 1159105