Abstract
Let E/F be a quadratic extension of number fields. We study periods and regularized periods of cusp forms and Eisenstein series on \(\operatorname {GL}_{n}( \mathbf {A}_{E})\) over a unitary group of a Hermitian form with respect to E/F. We provide factorization for these periods into locally defined functionals, express these factors in terms of suitably defined local periods and characterize global distinction. We also study in detail the analogous local question and analyze the space of invariant linear forms under a unitary group.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, 1989. MR 1007299 (90m:22041).
A. Aizenbud and D. Gourevitch, Smooth transfer of Kloosterman integrals (the Archimedean case), Amer. J. Math., to appear.
A. Aizenbud and D. Gourevitch, Schwartz functions on Nash manifolds, Int. Math. Res. Not., 5 (2008), rnm155. 37. MR 2418286 (2010g:46124).
A. Aizenbud and D. Gourevitch, Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis’s theorem, Duke Math. J., 149 (2009), 509–567, with an appendix by the authors and Eitan Sayag. MR 2553879 (2011c:22026).
A. Aizenbud and D. Gourevitch, Multiplicity one theorem for \((\mathrm{GL}_{n+1}(\Bbb{R}),\mathrm{GL}_{n}(\Bbb{R}))\), Sel. Math. New Ser., 15 (2009), 271–294. MR 2529937 (2010i:22012).
A. Aizenbud and D. Gourevitch, The de-Rham theorem and Shapiro lemma for Schwartz function on Nash manifolds, Isr. J. Math., 177 (2010), 155–188. MR 2684417.
A. Aizenbud, D. Gourevitch, and E. Sayag, (GL n+1(F),GL n (F)) is a Gelfand pair for any local field F, Compos. Math., 144 (2008), 1504–1524. MR 2474319 (2009k:22022).
J. G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in G(Q), Duke Math. J., 45 (1978), 911–952. MR 518111 (80d:10043).
J. Arthur, A measure on the unipotent variety, Can. J. Math., 37 (1985), 1237–1274. MR 828844 (87m:22049).
J. Arthur, On a family of distributions obtained from orbits, Can. J. Math., 38 (1986), 179–214. MR 835041 (87k:11058).
D. Barbasch, The unitary dual for complex classical Lie groups, Invent. Math., 96 (1989), 103–176. MR 981739 (90c:22044).
E. M. Baruch, A proof of Kirillov’s conjecture, Ann. Math. (2), 158 (2003), 207–252. MR 1999922 (2004f:22012).
J.-L. Brylinski and P. Delorme, Vecteurs distributions H-invariants pour les séries principales généralisées d’espaces symétriques réductifs et prolongement méromorphe d’intégrales d’Eisenstein, Invent. Math., 109 (1992), 619–664. MR 1176208 (93m:22016).
P. Blanc and P. Delorme, Vecteurs distributions H-invariants de représentations induites, pour un espace symétrique réductif p-adique G/H, Ann. Inst. Fourier (Grenoble), 58 (2008), 213–261. MR 2401221 (2009e:22015).
J. N. Bernstein, P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case), in Lie group representations, II. Lecture Notes in Math., vol. 1041, pp. 50–102, Springer, Berlin, 1984. MR 748505 (86b:22028).
J. Bernstein and B. Krötz, Smooth Fréchet globalizations of Harish-Chandra modules, Israel J. Math., to appear.
C. J. Bushnell and P. C. Kutzko, The Admissible Dual of GL(N) via Compact Open Subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, Princeton, 1993. MR 1204652 (94h:22007).
I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group GL(n,F), where F is a local non-Archimedean field, Usp. Mat. Nauk, 31 (1976), 5–70. MR 0425030 (54 #12988) no. 3(189).
I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive \(\mathfrak{p}\)-adic groups. I, Ann. Sci. Éc. Norm. Super. (4), 10 (1977), 441–472. MR 0579172 (58 #28310).
J. Carmona and P. Delorme, Base méromorphe de vecteurs distributions H-invariants pour les séries principales généralisées d’espaces symétriques réductifs: equation fonctionnelle, J. Funct. Anal., 122 (1994), 152–221. MR 1274587 (95g:22021).
N. B. Châu, Le lemme fondamental de Jacquet et Ye en caractéristique positive, Duke Math. J., 96 (1999), 473–520. MR 1671212 (2000f:11059).
G. Chinta and O. Offen, Unitary periods, Hermitian forms and points on flag varieties, Math. Ann., 339 (2007), 891–913. MR 2341906 (2009e:11070).
W. Casselman and J. Shalika, The unramified principal series of p-adic groups. II. The Whittaker function, Compos. Math., 41 (1980), 207–231. MR 581582 (83i:22027).
W. Casselman and F. Shahidi, On irreducibility of standard modules for generic representations, Ann. Sci. Éc. Norm. Super. (4), 31 (1998), 561–589. MR 1634020 (99f:22028).
P. Deligne, D. Kazhdan, and M.-F. Vignéras, Représentations des algèbres centrales simples p-adiques, in Representations of Reductive Groups over a Local Field, Travaux en Cours, pp. 33–117, Hermann, Paris, 1984. MR 771672 (86h:11044).
M. Flensted-Jensen, T. Ōshima, and H. Schlichtkrull, Boundedness of certain unitarizable Harish-Chandra modules, in Representations of Lie groups, Kyoto, Hiroshima, 1986. Adv. Stud. Pure Math., vol. 14, pp. 651–660, Academic Press, Boston, 1988. MR 1039855 (91e:22015).
I. M. Gel’fand and D. A. Kajdan, Representations of the group GL(n,K) where K is a local field, in Lie Groups and Their Representations, pp. 95–118, Halsted, New York, 1975. MR 0404534 (53 #8334).
R. Gow, Two multiplicity-free permutation representations of the general linear group GL(n,q 2), Math. Z., 188 (1984), 45–54. MR 767361 (86a:20008).
G. Henniart, Induction automorphe pour \(\mathrm{GL}(n,\Bbb{C})\), J. Funct. Anal., 258 (2010), 3082–3096. MR 2595735 (2011e:22028).
G. Henniart and R. Herb, Automorphic induction for GL(n) (over local non-Archimedean fields), Duke Math. J., 78 (1995), 131–192. MR 1328755 (96i:22038).
Y. Hironaka, Spherical functions and local densities on Hermitian forms, J. Math. Soc. Jpn., 51 (1999), 553–581. MR 1691493 (2000c:11064).
G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math., 366 (1986), 53–120. MR 833013 (87k:11066).
J. Hakim and Z. Mao, Supercuspidal representations of GL(n) distinguished by a unitary subgroup, Pac. J. Math., 185 (1998), 149–162. MR 1653208 (99j:22023).
J. Hakim and F. Murnaghan, Globalization of distinguished supercuspidal representations of GL(n), Can. Math. Bull., 45 (2002), 220–230. MR 1904086 (2003f:22022).
J. Hakim and F. Murnaghan, Tame supercuspidal representations of GL(n) distinguished by a unitary group, Compos. Math., 133 (2002), 199–244. MR 1923582 (2003g:22019).
A. G. Helminck and S. P. Wang, On rationality properties of involutions of reductive groups, Adv. Math., 99 (1993), 26–96. MR 1215304 (94d:20051).
A. Ichino, Trilinear forms and the central values of triple product L-functions, Duke Math. J., 145 (2008), 281–307. MR 2449948 (2009i:11066).
A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Funct. Anal., 19 (2010), 1378–1425. MR 2585578.
H. Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. Fr., 95 (1967), 243–309. MR 0271275 (42 #6158).
H. Jacquet, Relative Kloosterman integrals for GL(3). II, Can. J. Math., 44 (1992), 1220–1240. MR 1192415 (94c:11048).
H. Jacquet, The continuous spectrum of the relative trace formula for GL(3) over a quadratic extension, Isr. J. Math., 89 (1995), 1–59. MR 1324453 (96a:22029).
H. Jacquet, A theorem of density for Kloosterman integrals, Asian J. Math., 2 (1998), 759–778. Mikio Sato: a great Japanese mathematician of the twentieth century. MR 1734128 (2001e:11054).
H. Jacquet, Factorization of period integrals, J. Number Theory, 87 (2001), 109–143. MR 1816039 (2002a:11050).
H. Jacquet, Transfert lisse d’intégrales de Kloosterman, C. R. Math. Acad. Sci. Paris, 335 (2002), 229–232. MR 1933663 (2003k:11082).
H. Jacquet, Facteurs de transfert pour les intégrales de Kloosterman, C. R. Math. Acad. Sci. Paris, 336 (2003), 121–124. MR 1969564 (2004e:11052).
H. Jacquet, Smooth transfer of Kloosterman integrals, Duke Math. J., 120 (2003), 121–152. MR 2010736 (2005a:11066).
H. Jacquet, Integral representation of Whittaker functions, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 373–419, Johns Hopkins Univ. Press, Baltimore, 2004. MR 2058615 (2005f:11100).
H. Jacquet, Kloosterman identities over a quadratic extension, Ann. Math. (2), 160 (2004), 755–779. MR 2123938 (2006d:11051).
H. Jacquet, Kloosterman identities over a quadratic extension. II, Ann. Sci. Éc. Norm. Super. (4), 38 (2005), 609–669. MR 2172953 (2006j:11070).
H. Jacquet, Kloosterman integrals for \(\mathrm{GL}(2,\Bbb{R})\), Pure Appl. Math. Q., 1 (2005), 257–289. MR 2194725 (2007j:22019) no. 2, part 1.
H. Jacquet, Archimedean Rankin-Selberg integrals, in Automorphic Forms and L-functions II. Local Aspects, Contemp. Math., vol. 489, pp. 57–172, Amer. Math. Soc., Providence, 2009. MR 2533003 (2011a:11103).
H. Jacquet, Distinction by the quasi-split unitary group, Isr. J. Math., 178 (2010), 269–324. MR 2733072 (2011k:11073).
H. Jacquet, E. Lapid, and J. Rogawski, Periods of automorphic forms, J. Am. Math. Soc., 12 (1999), 173–240. MR 1625060 (99c:11056).
H. Jacquet, E. Lapid, and S. Rallis, A spectral identity for skew symmetric matrices, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 421–455, Johns Hopkins Univ. Press, Baltimore, 2004. MR 2058616 (2005h:11105).
H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Am. J. Math., 105 (1983), 367–464. MR 701565 (85g:11044).
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Am. J. Math., 103 (1981), 777–815. MR 623137 (82m:10050b).
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Am. J. Math., 103 (1981), 499–558. MR 618323 (82m:10050a).
H. Jacquet and J. Shalika, The Whittaker models of induced representations, Pac. J. Math., 109 (1983), 107–120. MR 716292 (85h:22023).
H. Jacquet and J. Shalika, Rankin-Selberg convolutions: Archimedean theory, in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I. Israel Math. Conf. Proc. (Ramat Aviv, 1989), vol. 2, pp. 125–207, Weizmann, Jerusalem, 1990. MR 1159102 (93d:22022).
H. Jacquet and Y. Ye, Une remarque sur le changement de base quadratique, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 671–676. MR 1081622 (92j:11046).
H. Jacquet and Y. Ye, Relative Kloosterman integrals for GL(3), Bull. Soc. Math. Fr., 120 (1992), 263–295. MR 1180831 (94c:11047).
H. Jacquet and Y. Ye, Distinguished representations and quadratic base change for GL(3), Trans. Am. Math. Soc., 348 (1996), 913–939. MR 1340178 (96h:11041).
H. Jacquet and Y. Ye, Germs of Kloosterman integrals for GL(3), Trans. Am. Math. Soc., 351 (1999), 1227–1255. MR 1443878 (99j:11053).
N. Lagier, Terme constant de fonctions sur un espace symétrique réductif p-adique, J. Funct. Anal., 254 (2008), 1088–1145. MR 2381204 (2009d:22013).
E. M. Lapid, On the fine spectral expansion of Jacquet’s relative trace formula, J. Inst. Math. Jussieu, 5 (2006), 263–308. MR 2225043 (2007d:11059).
E. M. Lapid, A remark on Eisenstein series, in Eisenstein Series and Applications. Progr. Math., vol. 258, pp. 239–249, Birkhäuser, Boston, 2008. MR 2402686 (2009m:11072).
J.-P. Labesse and R. P. Langlands, L-indistinguishability for SL(2), Can. J. Math., 31 (1979), 726–785. MR 540902 (81b:22017).
E. Lapid and A. Mínguez, On a determinantal formula of Tadić, Amer. J. Math., to appear.
E. Lapid and Z. Mao, On the asymptotics of Whittaker functions, Represent. Theory, 13 (2009), 63–81. MR 2495561 (2010b:22024).
E. Lapid and W. Müller, Spectral asymptotics for arithmetic quotients of \(\mathrm{SL}(n,\Bbb{R})/\mathrm{SO}(n)\), Duke Math. J., 149 (2009), 117–155. MR 2541128.
E. Lapid and O. Offen, Compact unitary periods, Compos. Math., 143 (2007), 323–338. MR 2309989 (2008g:11091).
E. Lapid and J. Rogawski, Stabilization of periods of Eisenstein series and Bessel distributions on GL(3) relative to U(3), Doc. Math., 5 (2000), 317–350 (electronic). MR 1767567 (2002b:11068).
E. Lapid and J. Rogawski, Periods of Eisenstein series, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 513–516. MR 1860921 (2002k:11072).
E. M. Lapid and J. D. Rogawski, Periods of Eisenstein series: the Galois case, Duke Math. J., 120 (2003), 153–226. MR 2010737 (2004m:11077).
C. Mœglin and J.-L. Waldspurger, Sur l’involution de Zelevinski, J. Reine Angew. Math., 372 (1986), 136–177. MR 863522 (88c:22019).
C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. Éc. Norm. Super. (4), 22 (1989), 605–674. MR 1026752 (91b:22028).
B. C. Ngô, Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye, Ann. Sci. Éc. Norm. Super. (4), 32 (1999), 619–679. MR 1710755 (2001g:11076).
O. Offen, Stable relative Bessel distributions on GL(n) over a quadratic extension, Am. J. Math., 129 (2007), 1183–1226. MR 2354318 (2009j:22021).
O. Offen, Unitary periods and Jacquet’s relative trace formula, in Automorphic Forms and L-functions I. Global Aspects, Contemp. Math., vol. 488, pp. 183–236, Amer. Math. Soc., Providence, 2009. MR 2522031 (2010k:11083).
N. Skovhus Poulsen, On C ∞-vectors and intertwining bilinear forms for representations of Lie groups, J. Funct. Anal., 9 (1972), 87–120. MR 0310137 (46 #9239).
D. Prasad, On a conjecture of Jacquet about distinguished representations of GL(n), Duke Math. J., 109 (2001), 67–78. MR 1844204 (2002g:22036).
P. Sarnak, A letter to Cathleen Morawetz, http://www.math.princeton.edu/sarnak (2004).
F. Shahidi, Fourier transforms of intertwining operators and Plancherel measures for GL(n), Am. J. Math., 106 (1984), 67–111. MR 729755 (86b:22031).
F. Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J., 52 (1985), 973–1007. MR 816396 (87m:11049).
F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. Math. (2), 132 (1990), 273–330. MR 1070599 (91m:11095).
T. Shintani, Two remarks on irreducible characters of finite general linear groups, J. Math. Soc. Jpn., 28 (1976), 396–414. MR 0414730 (54 #2825).
T. A. Springer, Some results on algebraic groups with involutions, in Algebraic Groups and Related Topics. Adv. Stud. Pure Math., vol. 6, pp. 525–543, North-Holland, Amsterdam, 1985. MR 803346 (86m:20050).
M. Tadić, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. Éc. Norm. Super. (4), 19 (1986), 335–382. MR 870688 (88b:22021).
E. van den Ban and P. Delorme, Quelques propriétés des représentations sphériques pour les espaces symétriques réductifs, J. Funct. Anal., 80 (1988), 284–307. MR 961900 (89j:22025).
J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math., 54 (1985), 173–242. MR 783511 (87g:11061b).
N. R. Wallach, Real Reductive Groups. II. Pure and Applied Mathematics, vol. 132, Academic Press, Boston, 1992. MR 1170566 (93m:22018).
A. Weil, Adeles and Algebraic Groups. Progress in Mathematics, vol. 23, Birkhäuser, Boston, 1982. With appendices by M. Demazure and Takashi Ono. MR 670072 (83m:10032).
Y. Ye, Kloosterman integrals and base change, in Number Theory and Its Applications in China. Contemp. Math., vol. 77, pp. 163–170, Amer. Math. Soc., Providence, 1988. MR 973234 (90b:11058).
Y. Ye, Kloosterman integrals and base change for GL(2), J. Reine Angew. Math., 400 (1989), 57–121. MR 1013725 (90i:11134).
Y. Ye, The fundamental lemma of a relative trace formula for GL(3), Compos. Math., 89 (1993), 121–162. MR 1255692 (95b:22023).
Y. Ye, An integral transform and its applications, Math. Ann., 300 (1994), 405–417. MR 1304430 (95j:11045).
Y. Ye, The lifting of Kloosterman sums, J. Number Theory, 51 (1995), 275–287. MR 1326749 (97a:11126).
Y. Ye, A Kloosterman sum in a relative trace formula for GL4, Represent. Theory, 2 (1998), 370–392 (electronic). MR 1641835 (99j:11092).
A. V. Zelevinsky, Induced representations of reductive \({\mathfrak{p}}\)-adic groups. II. On irreducible representations of GL(n), Ann. Sci. Éc. Norm. Super. (4), 13 (1980), 165–210. MR 584084 (83g:22012).
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Jonathan Rogawski
The second and third named authors were partially supported by grants from the Israel Science Foundation.
About this article
Cite this article
Feigon, B., Lapid, E. & Offen, O. On representations distinguished by unitary groups. Publ.math.IHES 115, 185–323 (2012). https://doi.org/10.1007/s10240-012-0040-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10240-012-0040-z