×

On extensions of supersingular representations of \(\mathrm{SL}_2(\mathbb{Q}_p)\). (English) Zbl 1477.22011

In this paper, the author put a great effort in the background material and this was really needed throughout. The main goal, see Theorem 1.1, is to calculate the dimensions of the spaces of degree one extensions \(\text{Ext}^1_{\mathrm{SL}_2(\mathbb{Q}_p)}\), for any two irreducible supersingular representations of \(\mathrm{SL}_2(\mathbb{Q}_p)\), when \(p>3 \). Extensions of irreducible supersingular representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\) were studied in [V. Paskunas, Astérisque 331, 317–353 (2010; Zbl 1204.22013)], and the dimensions of those spaces were calculated. For extensions between non-supersingular representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\) and in general for other classes of reductive groups see [C. Breuil and V. Paskunas, Mem. Amer. Math. Soc. 216 (1016) (2012; Zbl 1245.22010)] and [J. Hauseux, Algebra Number Theory 12 (4), 779–831 (2018; Zbl 1395.22008)], respectively.
Other than the introduction, the paper contains two sections with several subsections, each contributing a step in the proof of the main theorem. In the first section, Extensions of pro \(p\)-Iwahori Heche algebra modules were calculated from resolutions constructed for Hecke modules due to Schneider and Ollivier [R. Ollivier and P. Schneider, Adv. Math. 327, 52–127 (2018; Zbl 1427.20007)]. The author continues the work in the second section; first by considering the equivalence between the category of smooth representations of \(\mathrm{SL}_2(\mathbb{Q}_p)\), generated by the pro \(p\)-Iwahori invariant vectors, and the module category of the pro \(p\)-Iwahori Hecke algebra of \(\mathrm{SL}_2(\mathbb{Q}_p)\), refering to Theorem 5.2 in [K. Koziol Int. Math. Res. Not. IMRN (4), 1090–1125 (2016; Zbl 1347.11048)]. This equivalence between the above two categories, induces a certain Ext-spectral sequence which was then used to calculate the \(\text{Ext}^1_{\mathrm{SL}_2(\mathbb{Q}_p)}\).
Some applications of the main result are also derived. For example, to determine the structure of the pro \(p\)-Iwahori invariants of extensions of supersingular representations of \(\mathrm{SL}_2(\mathbb{Q}_p)\). Hence to get the explicit construction of those extensions of supersingular representations of \(\mathrm{SL}_2(\mathbb{Q}_p)\) generated by pro \(p \)-Iwahori invariants.
Finally, the paper is well organized, written neatly and in clear mathematical language.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abdellatif, Ramla, Classification des représentations modulo \(p\) de \(SL(2, F)\), Bull. Soc. Math. France, 142, 3, 537-589 (2014), MR 3295722 · Zbl 1319.11029
[2] Breuil, Christophe, Sur quelques représentations modulaires et \(p\)-adiques de \(GL_2(Q_p). I\), Compos. Math., 138, 2, 165-188 (2003), MR 2018825 · Zbl 1044.11041
[3] Breuil, Christophe; Paškūnas, Vytautas, Towards a modulo \(p\) Langlands correspondence for \(GL_2\), Mem. Amer. Math. Soc., 216, 1016 (2012), vi+114. MR 2931521 · Zbl 1245.22010
[4] Hauseux, Julien, Parabolic induction and extensions, Algebra Number Theory, 12, 4, 779-831 (2018), MR 3830204 · Zbl 1395.22008
[5] Kozioł, Karol, Pro-\(p\)-Iwahori invariants for \(SL_2\) and \(L\)-packets of Hecke modules, Int. Math. Res. Not. IMRN, 4, 1090-1125 (2016), MR 3493443 · Zbl 1347.11048
[6] Ollivier, Rachel; Schneider, Peter, Pro-\(p\) Iwahori-Hecke algebras are Gorenstein, J. Inst. Math. Jussieu, 13, 4, 753-809 (2014), MR 3249689 · Zbl 1342.20004
[7] Ollivier, Rachel; Schneider, Peter, A canonical torsion theory for pro-\(p\) Iwahori-Hecke modules, Adv. Math., 327, 52-127 (2018), MR 3761991 · Zbl 1427.20007
[8] Paškūnas, Vytautas, Extensions for supersingular representations of \(GL_2(Q_p)\), Astérisque, 331, 317-353 (2010), MR 2667891 · Zbl 1204.22013
[9] Paškūnas, Vytautas, The image of Colmez’s Montreal functor, Publ. Math. Inst. Hautes Études Sci., 118, 1-191 (2013), MR 3150248 · Zbl 1297.22021
[10] Paškūnas, Vytautas, Blocks for mod \(p\) Representations of \(GL_2(Q_p)\), London Math. Soc. Lecture Note Ser., vol. 415 (2014), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, MR 3444235 · Zbl 1365.11049
[11] Schneider, Peter, Smooth representations and Hecke modules in characteristic \(p\), Pacific J. Math., 279, 1-2, 447-464 (2015), MR 3437786 · Zbl 1359.16011
[12] Serre, Jean-Pierre, Galois Cohomology, Springer Monogr. Math. (2002), Springer-Verlag: Springer-Verlag Berlin, Translated from the French by Patrick Ion and revised by the author. MR 1867431 · Zbl 1004.12003
[13] Vignéras, Marie-France, Pro-\(p\)-Iwahori Hecke ring and supersingular \(\overline{F}_p\)-representations, Math. Ann., 331, 3, 523-556 (2005), MR 2122539 · Zbl 1107.22011
[14] Vigneras, Marie-France, The pro-p-iwahori hecke algebra of a reductive p-adic group iii (spherical hecke algebras and supersingular modules), J. Inst. Math. Jussieu, 1-38 (2015) · Zbl 1332.22019
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.