Extended Deligne-Lusztig varieties for general and special linear groups - Mathematics > Representation TheoryReport as inadecuate




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Abstract: We give a generalisation of Deligne-Lusztig varieties for general and speciallinear groups over finite quotients of the ring of integers in anon-archimedean local field. Previously, a generalisation was given by Lusztigby attaching certain varieties to unramified maximal tori inside Borelsubgroups. In this paper we associate a family of so-called extendedDeligne-Lusztig varieties to all tamely ramified maximal tori of the group.Moreover, we analyse the structure of various generalised Deligne-Lusztigvarieties, and show that the -unramified- varieties, including a certainnatural generalisation, do not produce all the irreducible representations ingeneral. On the other hand, we prove results which together with somecomputations of Lusztig show that for$\SL {2}\mathbb{F} {q}\varpi-\varpi^{2}$, with odd $q$, the extendedDeligne-Lusztig varieties do indeed afford all the irreducible representations.



Author: Alexander Stasinski

Source: https://arxiv.org/



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