Equivalence of categories between coefficient systems and systems of idempotents

Lanard, Thomas

doi:10.1090/ert/572

Equivalence of categories between coefficient systems and systems of idempotents
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by Thomas Lanard

Represent. Theory 25 (2021), 422-439

DOI: https://doi.org/10.1090/ert/572

Published electronically: June 2, 2021

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Abstract:

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $\operatorname {Rep}_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline {\mathbb {Z}}_{\ell }$, $\ell \neq p$, by Dat for $GL_{n}$ and the author for a more general group. Wang proved in the case of $GL_{n}$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $GL_{n}$ and a unipotent block of another group. In this paper, we generalize Wang’s equivalence of category to a connected reductive group on a non-archimedean local field.

References

F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923, DOI 10.1007/BF02715544
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
Jean-François Dat, Equivalences of tame blocks for $p$-adic linear groups, Math. Ann. 371 (2018), no. 1-2, 565–613. MR 3788858, DOI 10.1007/s00208-018-1648-1
Thomas Lanard, Sur les $\ell$-blocs de niveau zéro des groupes $p$-adiques, Compos. Math. 154 (2018), no. 7, 1473–1507 (French, with English summary). MR 3826462, DOI 10.1112/s0010437x18007133
Thomas Lanard, Sur les $\ell$-blocs de niveau zéro des groupes $p$-adiques II, arXiv e-prints (2018), arXiv:1806.09543.
Ralf Meyer and Maarten Solleveld, Resolutions for representations of reductive $p$-adic groups via their buildings, J. Reine Angew. Math. 647 (2010), 115–150. MR 2729360, DOI 10.1515/CRELLE.2010.075
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, DOI 10.1007/BF02699536
Marie-France Vignéras, Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics, vol. 137, Birkhäuser Boston, Inc., Boston, MA, 1996 (French, with English summary). MR 1395151
Marie-France Vignéras, Cohomology of sheaves on the building and $R$-representations, Invent. Math. 127 (1997), no. 2, 349–373. MR 1427623, DOI 10.1007/s002220050124
Haoran Wang, L’espace symétrique de Drinfeld et correspondance de Langlands locale II, Math. Ann. 369 (2017), no. 3-4, 1081–1130 (French, with English summary). MR 3713536, DOI 10.1007/s00208-016-1489-8