Let be a local non-archimedean field and its ring of integers. Let be a Bernstein component of the category of smooth representations of , let be a Bushnell–Kutzko -type, and let be the centre of the Bernstein component . Let be a direct summand of . We will begin by computing , where is the residue field at maximal ideal of , and the maximal ideal belongs to a Zariski-dense set in .
This result will allow us to deduce that the endomorphism ring is isomorphic to , when appears with multiplicity one in .
Soient un corps local non-archimédien et son anneau des entiers. Soit une composante de Bernstein de la catégorie des représentations lisses de Soient un -type de Bushnell–Kutzko et le centre de Bernstein de la composante . Soit un facteur direct de . Nous commençons par calculer , où est le corps résiduel de en un idéal maximal , et appartient à un ensemble Zariski dense dans .
Ce résultat nous permet ensuite de déduire que l’anneau des endomorphismes est isomorphe à , si apparait avec multiplicité un dans .
Revised:
Accepted:
Published online:
Classification: 22E50, 11F70
Keywords: smooth representations, -adic groups, Bernstein centre
Author's affiliations:
@article{JTNB_2020__32_1_49_0, author = {Alexandre Pyvovarov}, title = {The endomorphism ring of projectives and the {Bernstein} centre}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {49--71}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {32}, number = {1}, year = {2020}, doi = {10.5802/jtnb.1111}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1111/} }
TY - JOUR TI - The endomorphism ring of projectives and the Bernstein centre JO - Journal de Théorie des Nombres de Bordeaux PY - 2020 DA - 2020/// SP - 49 EP - 71 VL - 32 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1111/ UR - https://doi.org/10.5802/jtnb.1111 DO - 10.5802/jtnb.1111 LA - en ID - JTNB_2020__32_1_49_0 ER -
Alexandre Pyvovarov. The endomorphism ring of projectives and the Bernstein centre. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 49-71. doi : 10.5802/jtnb.1111. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1111/
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