## Duke Mathematical Journal

### From pro-$p$ Iwahori–Hecke modules to $(\varphi,\Gamma)$-modules, I

Elmar Grosse-Klönne

#### Abstract

Let ${\mathfrak{o}}$ be the ring of integers in a finite extension $K$ of ${\mathbb{Q}}_{p}$, and let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb{Q}}_{p}$, and let $T$ be a maximal split torus in $G$. Let ${\mathcal{H}}(G,I_{0})$ be the pro-$p$ Iwahori–Hecke ${\mathfrak{o}}$-algebra. Given a semi-infinite reduced chamber gallery (alcove walk) $C^{({\bullet})}$ in the $T$-stable apartment, a period $\phi\in N(T)$ of $C^{({\bullet})}$ of length $r$, and a homomorphism $\tau:{\mathbb{Z}}_{p}^{\times}\to T$ compatible with $\phi$, we construct a functor from the category $\operatorname{Mod}^{\mathrm{fin}}({\mathcal{H}}(G,I_{0}))$ of finite-length ${\mathcal{H}}(G,I_{0})$-modules to étale $(\varphi^{r},\Gamma)$-modules over Fontaine’s ring ${\mathcal{O}}_{\mathcal{E}}$. If $G=\operatorname{GL}_{d+1}({\mathbb{Q}}_{p})$, then there are essentially two choices of $(C^{({\bullet})},\phi,\tau)$ with $r=1$, both leading to a functor from $\operatorname{Mod}^{\mathrm{fin}}({\mathcal{H}}(G,I_{0}))$ to étale $(\varphi,\Gamma)$-modules and hence to $\mathrm{Gal}_{{\mathbb{Q}}_{p}}$-representations. Both induce a bijection between the set of absolutely simple supersingular ${\mathcal{H}}(G,I_{0})\otimes_{\mathfrak{o}}k$-modules of dimension $d+1$ and the set of irreducible representations of $\mathrm{Gal}_{{\mathbb{Q}}_{p}}$ over $k$ of dimension $d+1$. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of $G$ over $K$. For $d=1$, we recover Colmez’s functor (when restricted to ${\mathfrak{o}}$-torsion $\operatorname{GL}_{2}({\mathbb{Q}}_{p})$-representations generated by their pro-$p$ Iwahori invariants).

#### Article information

Source
Duke Math. J., Volume 165, Number 8 (2016), 1529-1595.

Dates
Revised: 17 July 2015
First available in Project Euclid: 25 February 2016

https://projecteuclid.org/euclid.dmj/1456412785

Digital Object Identifier
doi:10.1215/00127094-3450101

Mathematical Reviews number (MathSciNet)
MR3504178

Zentralblatt MATH identifier
1364.11103

#### Citation

Grosse-Klönne, Elmar. From pro- $p$ Iwahori–Hecke modules to $(\varphi,\Gamma)$ -modules, I. Duke Math. J. 165 (2016), no. 8, 1529--1595. doi:10.1215/00127094-3450101. https://projecteuclid.org/euclid.dmj/1456412785

#### References

• [1] P. N. Balister and S. Howson, Note on Nakayama’s lemma for compact $\Lambda$-modules, Asian J. Math. 1 (1997), 224–229.
• [2] L. Berger, On some modular representations of the Borel subgroup of $\operatorname{GL}_{2}({\mathbb{Q}}_{p})$, Compos. Math. 146 (2010), 58–80.
• [3] C. Breuil and F. Herzig, Ordinary representations of $G({\mathbb{Q}}_{p})$ and fundamental algebraic representations, Duke Math. J. 164 (2015), 1271–1352.
• [4] R. W. Carter and G. Lusztig, Modular representations of finite groups of Lie type, Proc. Lond. Math. Soc. (3) 32 (1976), 347–384.
• [5] P. Colmez, $(\phi,\Gamma)$-modules et représentations du mirabolique de $\operatorname{GL}_{2}({\mathbb{Q}}_{p})$, Astérisque 330 (2010), 61–153.
• [6] P. Colmez, Représentations de $\operatorname{GL}_{2}({\mathbb{Q}}_{p})$ et $(\phi,\Gamma)$-modules, Astérisque 330 (2010), 281–509.
• [7] M. Emerton, On a class of coherent rings, with applications to the smooth representation theory of $\operatorname{GL}_{2}(Q_{p})$ in characteristic $p$, preprint, 2008.
• [8] J.-M. Fontaine, “Représentations $p$-adiques des corps locaux, I” in The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Birkhäuser, Boston, 1990, 249–309.
• [9] E. Grosse-Klönne, $p$-torsion coefficient systems for $\operatorname{SL}_{2}({\mathbb{Q}}_{p})$ and $\operatorname{GL}_{2}({\mathbb{Q}}_{p})$, J. Algebra 342 (2011), 97–110.
• [10] E. Grosse-Klönne, Locally unitary principal series representations of $\operatorname{GL}_{d+1}(F)$, Münster J. Math. 7 (2014), 115–134.
• [11] E. Grosse-Klönne, From pro-$p$ Iwahori–Hecke modules to $(\varphi,\Gamma)$-modules, II, preprint.
• [12] R. Ollivier, Parabolic induction and Hecke modules in characteristic $p$ for $p$-adic $\operatorname{GL}_{n}$, Algebra Number Theory 4 (2010), 701–742.
• [13] R. Ollivier and P. Schneider, Pro-$p$ Iwahori–Hecke algebras are Gorenstein, J. Inst. Math. Jussieu 13 (2014), 753–809.
• [14] R. Ollivier and V. Sécherre, Modules universels de $\operatorname{GL}_{3}$ sur un corps $p$-adique en caractéristique $p$, preprint, arXiv:1105.2957v1 [math.RT].
• [15] M. Vienney, Construction de $(\varphi,\Gamma)$-modules en caractéristique $p$, Ph.D. dissertation, École normale supérieure de Lyon, Lyon, France, 2012.
• [16] M.-F. Vignéras, Pro-$p$-Iwahori Hecke ring and supersingular $\overline{\mathbb{F}}_{p}$-representations, Math. Ann. 331 (2005), 523–556. Erratum, Math. Ann. 333 (2005), 699–701.