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DTSTAMP:20261021T070353
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SUMMARY:Workshop / Seminar:GLNT: The Galois module structure of Deligne--Lusztig varieties for $G=\mathrm{GL}_3$
DESCRIPTION:Abstract: Deligne and Lusztig defined a finite \'etale cover \(Y(w) \to X(w)\) of varieties which is attached to any connected reductive group \(\mathbf{G}\) defined over a finite field \( \mathbb{F}_q \)\, with an isogeny \(F\) whose some positive power is a Frobenius endomorphism\, and a Weyl group element \(w \in W\). We consider the  ramified Galois covers  \( \overline{Y}(w)\rightarrow  \overline{X}(w) \) of the natural compactifications of these Deligne-Lusztig varieties. Our goal is to calculate the Lefschetz numbers for the action of the covering group\, which is given by the points \(\mathbf{T}^{wF}\) of a torus with values in \(\mathbb{F}_p\)\, on the coherent cohomology of the structure sheaf of \( \overline{Y}(w)\). \nThe full collection of Lefschetz numbers\ndetermine the virtual representation of \(\mathbf T^{wF}\) on the\nalternating sum of these cohomology groups.  In the two cases of Deligne--Lusztig surfaces of type \(A_2\) and \(^{2}A_2\) for \( \mathbf{G} = \mathrm{GL}_3 \)\, we give explicit formulas for the Lefschetz numbers for a generator of the stabilizer of an irreducible component of \(\overline{Y}(w)\) .\nIt follows that the alternating coherent cohomology class of the full compactification \(\overline{Y}(w)\) is not virtually free\nover the full covering group.
UID:153898-21916610@events.umich.edu
URL:https://events.umich.edu/event/153898
CLASS:PUBLIC
STATUS:CONFIRMED
CATEGORIES:Mathematics,Number Theory
LOCATION:East Hall - 4088
CONTACT:
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