The endomorphism ring of projectives and the Bernstein centre

Pyvovarov, Alexandre

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 49-71.

Soient $F$ un corps local non-archimédien et ${𝒪}_F$ son anneau des entiers. Soit $\Omega$ une composante de Bernstein de la catégorie des représentations lisses de ${\mathrm{GL}}_n\left(F\right).$ Soient $(J,\lambda )$ un $\Omega$-type de Bushnell–Kutzko et ${ℨ}_{\Omega }$ le centre de Bernstein de la composante $\Omega$. Soit $\sigma$ un facteur direct de ${\mathrm{Ind}}_J^{{\mathrm{GL}}_n\left({𝒪}_F\right)}\lambda$. Nous commençons par calculer $\mathrm{c}\text{--}{\mathrm{Ind}}_{{\mathrm{GL}}_n\left({𝒪}_F\right)}^{{\mathrm{GL}}_n\left(F\right)}\sigma {\otimes }_{{ℨ}_{\Omega }}\kappa \left(𝔪\right)$, où $\kappa \left(𝔪\right)$ est le corps résiduel de ${ℨ}_{\Omega }$ en un idéal maximal $𝔪$, et $𝔪$ appartient à un ensemble Zariski dense dans $\mathrm{Spec}{ℨ}_{\Omega }$.

Ce résultat nous permet ensuite de déduire que l’anneau des endomorphismes ${\mathrm{End}}_{{\mathrm{GL}}_n\left(F\right)}\left(\mathrm{c}\text{--}{\mathrm{Ind}}_{{\mathrm{GL}}_n\left({𝒪}_F\right)}^{{\mathrm{GL}}_n\left(F\right)}\sigma \right)$ est isomorphe à ${ℨ}_{\Omega }$, si $\sigma$ apparait avec multiplicité un dans ${\mathrm{Ind}}_J^{{\mathrm{GL}}_n\left({𝒪}_F\right)}\lambda$.

Let $F$ be a local non-archimedean field and ${𝒪}_F$ its ring of integers. Let $\Omega$ be a Bernstein component of the category of smooth representations of ${\mathrm{GL}}_n\left(F\right)$, let $(J,\lambda )$ be a Bushnell–Kutzko $\Omega$-type, and let ${ℨ}_{\Omega }$ be the centre of the Bernstein component $\Omega$. Let $\sigma$ be a direct summand of ${\mathrm{Ind}}_J^{{\mathrm{GL}}_n\left({𝒪}_F\right)}\lambda$. We will begin by computing $\mathrm{c}\text{--}{\mathrm{Ind}}_{{\mathrm{GL}}_n\left({𝒪}_F\right)}^{{\mathrm{GL}}_n\left(F\right)}\sigma {\otimes }_{{ℨ}_{\Omega }}\kappa \left(𝔪\right)$, where $\kappa \left(𝔪\right)$ is the residue field at maximal ideal $𝔪$ of ${ℨ}_{\Omega }$, and the maximal ideal $𝔪$ belongs to a Zariski-dense set in $\mathrm{Spec}{ℨ}_{\Omega }$.

This result will allow us to deduce that the endomorphism ring ${\mathrm{End}}_{{\mathrm{GL}}_n\left(F\right)}\left(\mathrm{c}\text{--}{\mathrm{Ind}}_{{\mathrm{GL}}_n\left({𝒪}_F\right)}^{{\mathrm{GL}}_n\left(F\right)}\sigma \right)$ is isomorphic to ${ℨ}_{\Omega }$, when $\sigma$ appears with multiplicity one in ${\mathrm{Ind}}_J^{{\mathrm{GL}}_n\left({𝒪}_F\right)}\lambda$.

Classification : 22E50, 11F70
Mots clés : smooth representations, $p$-adic groups, Bernstein centre

[1] Bass, Hyman Algebraic $K$-theory, Mathematics Lecture Note Series, W. A. Benjamin, 1968, xx+762 pages | MR | Zbl

[2] Bernstein, Joseph N. Le “centre” de Bernstein, Representations of reductive groups over a local field (Travaux en Cours), Hermann, 1984, pp. 1-32 | MR | Zbl

[3] Bourbaki, Nicolas Éléments de mathématique. Algèbre commutative. Chapitres 1 à 4, Masson, 1985, 362 pages | MR | Zbl

[4] Bourbaki, Nicolas Éléments de mathématique. Algèbre commutative. Chapitres 5 à 7, Masson, 1985, 351 pages | MR | Zbl

[5] Bourbaki, Nicolas Algebra II. Chapters 4–7, Elements of Mathematics, Springer, 2003, viii+461 pages (translated from the 1981 French edition by P. M. Cohn and J. Howie, Reprint of the 1990 English edition) | DOI | MR | Zbl

[6] Bourbaki, Nicolas Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples, Springer, 2012, x+489 pages (second revised edition of the 1958 edition) | DOI | MR | Zbl

[7] Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of $\mathrm{GL}\left(N\right)$ via compact open subgroups, Annals of Mathematics Studies, 129, Princeton University Press, 1993, xii+313 pages | DOI | MR | Zbl

[8] Bushnell, Colin J.; Kutzko, Philip C. Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. Lond. Math. Soc., Volume 77 (1998) no. 3, pp. 582-634 | DOI | MR | Zbl

[9] Bushnell, Colin J.; Kutzko, Philip C. Semisimple types in ${\mathrm{GL}}_n$, Compos. Math., Volume 119 (1999) no. 1, pp. 53-97 | DOI | MR | Zbl

[10] Caraiani, Ana; Emerton, Matthew; Gee, Toby; Geraghty, David; Paškūnas, Vytautas; Shin, Sug Woo Patching and the $p$-adic local Langlands correspondence, Camb. J. Math., Volume 4 (2016) no. 2, pp. 197-287 | DOI | MR | Zbl

[11] Dat, Jean-François Caractères à valeurs dans le centre de Bernstein, J. Reine Angew. Math., Volume 508 (1999), pp. 61-83 | DOI | MR | Zbl

[12] Dat, Jean-François Types et inductions pour les représentations modulaires des groupes $p$-adiques, Ann. Sci. Éc. Norm. Supér., Volume 32 (1999) no. 1, pp. 1-38 (with an appendix by Marie-France Vignéras) | DOI | Numdam | MR | Zbl

[13] Helm, David The Bernstein center of the category of smooth $W\left(k\right)\left[{\mathrm{GL}}_n\left(F\right)\right]$-modules, Forum Math. Sigma, Volume 4 (2016), e11, 98 pages | DOI | MR | Zbl

[14] Jantzen, Chris On the Iwahori-Matsumoto involution and applications, Ann. Sci. Éc. Norm. Supér., Volume 28 (1995) no. 5, pp. 527-547 | DOI | Numdam | MR | Zbl

[15] Pyvovarov, Alexandre On the Breuil-Schneider conjecture: Generic case (2018) (https://arxiv.org/abs/1803.01610)

[16] Renard, David Représentations des groupes réductifs $p$-adiques, Contributions in Mathematical and Computational Sciences, 17, Société Mathématique de France, 2010, vi+332 pages | MR | Zbl

[17] Schneider, Peter; Zink, Ernst-Wilhelm $K$-types for the tempered components of a $p$-adic general linear group, J. Reine Angew. Math., Volume 517 (1999), pp. 161-208 (with an appendix by Schneider and U. Stuhler) | DOI | MR | Zbl

[18] Swan, Richard G. Projective modules over Laurent polynomial rings, Trans. Am. Math. Soc., Volume 237 (1978), pp. 111-120 | DOI | MR | Zbl

[19] Zelevinsky, Andreĭ V. Induced representations of reductive $𝔭$-adic groups. II. On irreducible representations of $\mathrm{GL}\left(n\right)$, Ann. Sci. Éc. Norm. Supér., Volume 13 (1980) no. 2, pp. 165-210 | DOI | Numdam | MR | Zbl