×

A Miyawaki type lift for \(\mathrm{GSpin}(2,10)\). (English) Zbl 1430.11065

In the paper under the review, the authors provide a Miyawaki type lift for group \(\mathrm{GSpin}(2,10)\).
In their previous work [Compos. Math. 152, No. 2, 223–254 (2016; Zbl 1337.11031)], the authors have constructed holomorphic cusp forms on the Hermitian symmetric domain of the exceptional Lie group of type \(E_{7,3}\) over \(\mathbb{Q}\) from elliptic cusp forms with respect to \(\mathrm{SL}_2(\mathbb{Z})\).
Using such cusp forms, and assuming the existence of the Jacquet-Langlands correspondence that associates to automorphic representations of \(\mathrm{PGSpin}(2,10)(\mathbb{A})\) the automorphic representations of its split inner form \(\mathrm{PGSpin}(6,6)(\mathbb{A})\), authors obtain a construction of the holomorphic cusp forms on the Hermitian symmetric domain of the group \(\mathrm{Spin}(2,10)\).
Constructed cusp forms are similar to the Miyawaki lift obtained by T. Ikeda [Duke Math. J. 131, No. 3, 469–497 (2006; Zbl 1112.11022)] in the context of symplectic groups.

MSC:

11F55 Other groups and their modular and automorphic forms (several variables)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
20G41 Exceptional groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arthur, J.: The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups. American Mathematical Society Colloquium Publications, 61. American Mathematical Society, Providence, RI (2013) · Zbl 1310.22014
[2] Asgari, M., Raghuram, A.: A cuspidality criterion for the exterior square transfer of cusp forms on \[GL(4)\] GL(4). In: Clay Mathematics Proceedings, vol. 13, pp. 33-53, American Mathematical Society, Providence, RI (2011) · Zbl 1275.11086
[3] Asgari, M., Shahidi, F.: Generic transfer for general spin groups. Duke Math. J. 132, 137-190 (2006) · Zbl 1099.11028 · doi:10.1215/S0012-7094-06-13214-3
[4] Baily Jr., W.L.: An exceptional arithmetic group and its Eisenstein series. Ann. Math. 91, 512-549 (1970) · Zbl 0202.07901 · doi:10.2307/1970636
[5] Baily Jr., W.L., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. 84, 442-528 (1966) · Zbl 0154.08602 · doi:10.2307/1970457
[6] Borel, A., Jacquet, H.: Automorphic forms and automorphic representations. In: Proceedings of Symposia in Pure Mathematics, XXXIII, Automorphic forms, representations and L-functions, Part 1, pp. 189-207, AMS (1979) · Zbl 0414.22020
[7] Bourbaki, N.: Groupes et Algèbres de Lie, Chapitres 4,5 et 6, Paris (1981) · Zbl 0483.22001
[8] Chenevier, G., Renard, D.: Level one algebraic cusp forms of classical groups of small rank. Memoirs of the American Mathematical Society, vol. 237, American Mathematical Society, Providence (2015) · Zbl 1376.11036
[9] Demazure, M., Grothendieck, A. (Eds): Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1 (Lecture Notes in Mathematics 151) · Zbl 0209.24201
[10] Du, Z.: On the spectrum of \[(Spin(10,2), SL(2,\mathbb{R}))\](Spin(10,2),SL(2,R)) in \[E_{7,3}\] E7,3. Proc. AMS 139, 757-768 (2011) · Zbl 1218.22009 · doi:10.1090/S0002-9939-2010-10664-0
[11] Eie, M.: Jacobi-Eisenstein series of degree two over Cayley numbers. Rev. Mat. Iberoam. 16, 571-596 (2000) · Zbl 1009.11030 · doi:10.4171/RMI/284
[12] Eie, M., Krieg, A.: The Maass space on the half-plane of Cayley numbers of degree two. Math. Z. 210(1), 113-128 (1992) · Zbl 0729.11022 · doi:10.1007/BF02571786
[13] Fulton, W., Harris, J.: Representation Theory, A First Course. Springer, New York (1991) · Zbl 0744.22001
[14] Garrett, P.-B.: Pullbacks of Eisenstein series; applications. Automorphic forms of several variables (Katata, 1983). Progress in Mathematics, vol. 46, pp. 114-137, Birkhäuser Boston, Boston (1984) · Zbl 1337.11031
[15] Ginzburg, D.: On standard L-functions for \[E_6\] E6 and \[E_7\] E7. J. Reine Angew. Math. 465, 101-131 (1995) · Zbl 0832.11021
[16] Gross, B.: Groups over \[\mathbb{Z}\] Z. Invent. Math. 124(1-3), 263-279 (1996) · Zbl 0846.20049 · doi:10.1007/s002220050053
[17] Gross, B., Savin, G.: Motives with Galois group of type \[G_2\] G2: an exceptional theta-correspondence. Comput. Math. 114, 153-217 (1998) · Zbl 0931.11015
[18] Helminck, A.G., Wang, P.: On rationality properties of involutions of reductive groups. Adv. Math. 99(1), 26-96 (1993) · Zbl 0788.22022 · doi:10.1006/aima.1993.1019
[19] Ikeda, T.: On the lifting of elliptic cusp forms to Siegel cusp forms of degree \[2n2\] n. Ann. Math. 154, 641-681 (2001) · Zbl 0998.11023 · doi:10.2307/3062143
[20] Ikeda, T.: Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture. Duke Math. J. 131, 469-497 (2006) · Zbl 1112.11022 · doi:10.1215/S0012-7094-06-13133-2
[21] Jacquet, H., Shalika, J.: On Euler products and the classification of automorphic representations I, II. Am. J. Math. 103(499-558), 777-815 (1981) · Zbl 0491.10020 · doi:10.2307/2374050
[22] Kim, H.: Exceptional modular form of weight 4 on an exceptional domain contained in \[{\mathbb{C}}^{27}\] C27. Rev. Mat. Iberoam. 9, 139-200 (1993) · Zbl 0777.11015 · doi:10.4171/RMI/134
[23] Kim, H.: On local \[L\] L-functions and normalized intertwining operators. Can. J. Math. 57, 535-597 (2005) · Zbl 1096.11019 · doi:10.4153/CJM-2005-023-x
[24] Kim, H., Yamauchi, T.: Cusp forms for exceptional group of type \[E_7\] E7. Compos. Math. 152(2), 223-254 (2016) · Zbl 1337.11031 · doi:10.1112/S0010437X15007538
[25] Knapp, A.W.: Representation Theory of Semisimple Groups. Princeton University Press, Princeton (1986) · Zbl 0604.22001 · doi:10.1515/9781400883974
[26] Kitaoka, Y.: Arithmetic of quadratic forms. Cambridge Tracts in Mathematics, vol. 106. Cambridge University Press, Cambridge (1993) · Zbl 0785.11021
[27] Lawther, R.: Double cosets involving involutions in algebraic groups. Proc. Lond. Math. Soc. 70, 115-145 (1995) · Zbl 0836.20067 · doi:10.1112/plms/s3-70.1.115
[28] Lawther, R.: e-mail correspondences · Zbl 0491.10020
[29] Li, J.S.: Some results on the unramified principal series of \[p\] p-adic groups. Math. Ann. 292, 747-761 (1992) · Zbl 0804.22007 · doi:10.1007/BF01444646
[30] Luzgarev, A., Vavilov, N.A.: A Chevalley group of type \[E_7\] E7 in the 56-dimensional representation. (Russian. English, Russian summary) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 386 (2011), Voprosy Teorii Predstvalenii Algebr i Grupp. 20, 5-99, 288; translation in J. Math. Sci. 180(3), 197-251 (2012) · Zbl 0788.22022
[31] Onishchik, A.L., Vinberg, E.B.: Lie Groups and Lie Algebras III, Encyclopedia of Mathematical Sciences, vol. 41. Springer, Berlin (1990) · Zbl 0722.22004
[32] Platonov, V., Rapinchuk, A.: Algebraic Groups and Number Theory. Academic Press, Inc, Boston (1994) · Zbl 0841.20046
[33] Ramakrishnan, D.: Modularity of the Rankin-Selberg L-series, and multiplicity one for \[SL(2)\] SL(2). Ann. Math. 152(1), 45-111 (2000) · Zbl 0989.11023 · doi:10.2307/2661379
[34] Satake, I.: Algebraic structures of symmetric domains. In: Kanô Memorial Lectures, 4. Iwanami Shoten, Tokyo; Princeton University Press, Princeton (1980) · Zbl 0483.32017
[35] Sugano, T.: On Dirichlet series attached to holomorphic cusp fors on \[SO(2, q)\] SO(2,q). Automorphic forms and number theory (Sendai, 1983). Adv. Stud. Pure Math. 7, 333-362 (1985) · Zbl 0602.10018
[36] Tadic, M.: Spherical unitary dual of general linear group over non-Archimedean local field. Ann. Inst. Fourier 36(2), 47-55 (1986) · Zbl 0554.20009 · doi:10.5802/aif.1046
[37] Tsao, L.: The rationality of the Fourier coefficients of certain Eisenstein series on tube domains. Compos. Math. 32, 225-291 (1976) · Zbl 0346.10014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.