The first pro-𝑝-Iwahori cohomology of mod-𝑝 principal series for 𝑝-adic 𝐺𝐿_{𝑛}

Kozioł, Karol

doi:10.1090/tran/7619

The first pro-$p$-Iwahori cohomology of mod-$p$ principal series for $p$-adic $\mathrm {GL}_n$
HTML articles powered by AMS MathViewer

by Karol Kozioł PDF

Trans. Amer. Math. Soc. 372 (2019), 1237-1288 Request permission

Abstract:

Let $p\geq 3$ be a prime number and $F$ a $p$-adic field. Let $I_1$ denote the pro-$p$-Iwahori subgroup of $\mathrm {GL}_n(F)$, and let $\mathcal {H}$ denote the pro-$p$-Iwahori–Hecke algebra of $\mathrm {GL}_n(F)$ with respect to $I_1$ (over a coefficient field of characteristic $p$). We compute the structure of $\mathrm {H}^1(I_1,\pi )$ as an $\mathcal {H}$-module, where $\pi$ is a mod-$p$ principal series representation of $\mathrm {GL}_n(F)$. We also give some partial results about the structure of $\mathrm {H}^1(I_1,\pi )$ for a general split reductive group with an irreducible root system.

References

N. Abe, Parabolic inductions for pro-$p$-Iwahori Hecke algebras, ArXiv (2016).
N. Abe, Extension between simple modules of pro-$p$-Iwahori Hecke algebras, ArXiv (2017).
N. Abe, G. Henniart, F. Herzig, and M.-F. Vignéras, A classification of irreducible admissible $\textrm {mod}\, p$ representations of $p$-adic reductive groups, J. Amer. Math. Soc. 30 (2017), no. 2, 495–559. MR 3600042, DOI 10.1090/jams/862
Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
Christophe Breuil and Vytautas Paškūnas, Towards a modulo $p$ Langlands correspondence for $\textrm {GL}_2$, Mem. Amer. Math. Soc. 216 (2012), no. 1016, vi+114. MR 2931521, DOI 10.1090/S0065-9266-2011-00623-4
Christophe Breuil, The emerging $p$-adic Langlands programme, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 203–230. MR 2827792
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
Matthew Emerton, Ordinary parts of admissible representations of $p$-adic reductive groups II. Derived functors, Astérisque 331 (2010), 403–459 (English, with English and French summaries). MR 2667883
Matthew Emerton and Vytautas Paškūnas, On the effaceability of certain $\delta$-functors, Astérisque 331 (2010), 461–469 (English, with English and French summaries). MR 2667892
T. Gee, F. Herzig, and D. Savitt, General Serre weight conjectures, to appear in J. Eur. Math. Soc.
Michael Harris, Speculations on the $\textrm {mod}\, p$ representation theory of $p$-adic groups, Ann. Fac. Sci. Toulouse Math. (6) 25 (2016), no. 2-3, 403–418 (English, with English and French summaries). MR 3530163, DOI 10.5802/afst.1499
Julien Hauseux, Parabolic induction and extensions, Algebra Number Theory 12 (2018), no. 4, 779–831. MR 3830204, DOI 10.2140/ant.2018.12.779
Florian Herzig, The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations, Duke Math. J. 149 (2009), no. 1, 37–116. MR 2541127, DOI 10.1215/00127094-2009-036
Yongquan Hu, An application of a theorem of Emerton to $\textrm {mod}\,p$ representations of $\rm GL_2$, J. Lond. Math. Soc. (2) 96 (2017), no. 3, 545–564. MR 3742432, DOI 10.1112/jlms.12080
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR 185016
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
K. Kozioł, Hecke module structure on first and top pro-$p$-Iwahori cohomology, to appear in Acta Arith.
K. Kozioł, Homological dimension of simple pro-p-Iwahori–Hecke modules, ArXiv (2015).
Karol Kozioł, Pro-$p$-Iwahori invariants for $\textrm {SL}_2$ and $L$-packets of Hecke modules, Int. Math. Res. Not. IMRN 4 (2016), 1090–1125. MR 3493443, DOI 10.1093/imrn/rnv162
Michio Kuga, Walter Parry, and Chih Han Sah, Group cohomology and Hecke operators, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 223–266. MR 642860
Min Ho Lee, Hecke operators on cohomology, Rev. Un. Mat. Argentina 50 (2009), no. 1, 99–144. MR 2643521
Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
R. Ollivier and M.-F. Vigneras, Parabolic induction in characteristic p, ArXiv (2017).
Vytautas Paškūnas, Extensions for supersingular representations of $\textrm {GL}_2(\Bbb Q_p)$, Astérisque 331 (2010), 317–353 (English, with English and French summaries). MR 2667891
Vytautas Paškūnas, The image of Colmez’s Montreal functor, Publ. Math. Inst. Hautes Études Sci. 118 (2013), 1–191. MR 3150248, DOI 10.1007/s10240-013-0049-y
Y. H. Rhie and G. Whaples, Hecke operators in cohomology of groups, J. Math. Soc. Japan 22 (1970), 431–442. MR 285620, DOI 10.2969/jmsj/02240431
Peter Schneider, Smooth representations and Hecke modules in characteristic $p$, Pacific J. Math. 279 (2015), no. 1-2, 447–464. MR 3437786, DOI 10.2140/pjm.2015.279.447
Jean-Pierre Serre, Galois cohomology, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. MR 1867431
T. A. Springer, Linear algebraic groups, 2nd ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009. MR 2458469
Peter Schneider and Ulrich Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Inst. Hautes Études Sci. Publ. Math. 85 (1997), 97–191. MR 1471867
Marie-France Vigneras, The pro-$p$-Iwahori Hecke algebra of a reductive $p$-adic group I, Compos. Math. 152 (2016), no. 4, 693–753. MR 3484112, DOI 10.1112/S0010437X15007666
Marie-France Vigneras, The pro-$p$-Iwahori Hecke algebra of a reductive $p$-adic group III (spherical Hecke algebras and supersingular modules), J. Inst. Math. Jussieu 16 (2017), no. 3, 571–608. MR 3646282, DOI 10.1017/S1474748015000146