An explicit computation of $p$-stabilized vectors

MIYAUCHI, Michitaka; YAMAUCHI, Takuya

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 531-558.

Nous donnons une méthode concrète pour calculer les vecteurs $p$-stables dans l’espace des éléments fixés par un sous-groupe parahorique d’un groupe réductif $p$-adique. Nous discutons d’une application globale et, en particulier, nous donnons un exemple explicite d’un relèvement de Saito-Kurokawa $p$-stable.

In this paper, we give a concrete method to compute $p$-stabilized vectors in the space of parahori-fixed vectors for connected reductive groups over $p$-adic fields. An application to the global setting is also discussed. In particular, we give an explicit $p$-stabilized form of a Saito-Kurokawa lift.

[1] A. N. Andrianov, Quadratic forms and Hecke operators, Springer Berlin (1987). | MR | Zbl

[2] S. Böcherer, Siegfried On the Hecke operator $U\left(p\right)$. With an appendix by Ralf Schmidt. J. Math. Kyoto Univ. 45, 4 (2005), 807–829. | MR | Zbl

[3] A. Borel, Automorphic L -functions. Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I.,(1979) 27–61. | MR | Zbl

[4] A. Borel and H. Jacquet, Automorphic forms and automorphic representations. With a supplement “On the notion of an automorphic representation” by R. P. Langlands. Proc. Sympos. Pure Math., XXXIII, Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, 189–207. | MR | Zbl

[5] C. J. Bushnell and P. C. Kutzko, Smooth representations of reductive p -adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. | MR | Zbl

[6] W. Casselman, Introduction to admissible representations of $p$-adic groups, available at his homepage.

[7] Robert F. Coleman, Classical and overconvergent modular forms. Invent. Math. 124, 1-3 (1996), 215–241. | MR | Zbl

[8] P. Garrett, Representations with Iwahori-fixed vectors. Note available at his homepage http://www.math.umn.edu/~garrett/m/v/.

[9] M. Goresky, Compactifications and cohomology of modular varieties. Harmonic analysis, the trace formula, and Shimura varieties, , Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, (2005), 551–582. | MR | Zbl

[10] R. Howe Harish-Chandra homomorphisms for $p$-adic groups, 59 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, (1985). With the collaboration of A. Moy. | MR | Zbl

[11] T. Ibukiyama, Saito-Kurokawa liftings of level N and practical construction of Jacobi forms, Kyoto J. Math. 52, 1 (2012), 141–178. | MR | Zbl

[12] H. Jacquet, Sur les représentations des groupes réductifs $p$-adiques, C. R. Acad. Sci. Paris Ser. A-B 280 (1975), Aii, A1271–A1272. | MR | Zbl

[13] D. Keys, Principal series representations of special unitary groups over local fields, Compositio Math. 51, 1 (1984), 115–130. | Numdam | MR | Zbl

[14] B. Mazur, An “infinite fern” in the universal deformation space of Galois representations, Collect. Math., 48, 1-2 (1997), 155–193. Journées Arithmétiques (Barcelona, 1995). | MR | Zbl

[15] J. S. Milne, Introduction to Shimura varieties. In Harmonic analysis, the trace formula, and Shimura varieties, 4 of Clay Math. Proc., (2005) 265–378. Amer. Math. Soc., Providence, RI. | MR | Zbl

[16] B. Roberts and R. Schmidt, Local newforms for $GSp\left(4\right)$. Lecture Notes in Mathematics, 1918, Springer, Berlin, (2007), viii+307 pp. | MR | Zbl

[17] R. Salvati Manni and J. Top, Cusp forms of weight 2 for the group $\Gamma (4,8)$, Amer. J. Math. 115, (1993), 455–486. | MR | Zbl

[18] I. Satake, Algebraic structures of symmetric domains Kano Memorial Lectures, 4. Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., (1980). xvi+321 pp. | MR | Zbl

[19] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Reprint of the 1971 original. Publications of the Mathematical Society of Japan, 11, Kano Memorial Lectures, 1. Princeton University Press, Princeton, NJ, (1994). | MR | Zbl

[20] C. Skinner and E. Urban, Sur les déformations $p$-adiques de certaines représentations automorphes. J. Inst. Math. Jussieu 5, 4 (2006), 629–698. | MR | Zbl

[21] J. Tilouine, Nearly ordinary rank four Galois representations and $p$-adic Siegel modular forms. With an appendix by Don Blasius. Compos. Math. 142, 5 (2006), 1122–1156. | MR | Zbl