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Convex elements and cohomology of deep level Deligne-Lusztig varieties
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abstract
We essentially complete a program initiated by Boyarchenko--Weinstein to give a full description of the cohomology of deep level Deligne--Lusztig varieties for elliptic tori, with coefficients in arbitrary non-defining characteristics. We give several applications of our results: we show that the $\phi$-weight part of the cohomology is very often concentrated in a single degree, and is induced from a Yu-type subgroup. Also, we give applications to a previous work of the second author on decomposition of deep level Deligne--Lusztig representations, and to Feng's explicit construction of Fargues--Scholze parameters. Furthermore, a conjecture of Chan--Oi about the Drinfeld stratification is verified as a special case from our results.
Forward citations
Cited by 2 Pith papers
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Green functions for positive-depth Deligne--Lusztig induction
For large q, positive-depth Deligne-Lusztig induction matches the Yu-Kaletha-FKS algebraic construction for all Howe-unramified elliptic pairs, via a characterization theorem and a Green function comparison.
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An explicit decomposition of higher Deligne-Lsuztig representations
For q ≥ c_Λ with 2 ≤ c_Λ ≤ 4, the geometric representation κ_Λ from cohomology of higher Deligne-Lusztig varieties equals κ(Λ)⊗ϵ_Λ, giving an explicit irreducible decomposition of elliptic higher Deligne-Lusztig repre...
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