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 representations.
The cohomology of $p$-adic Deligne-Luszitg schemes of Coxeter type
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We determine the cohomology of the closed Drinfeld stratum of $p$-Deligne--Lusztig schemes of Coxeter type attached to arbitrary inner forms of unramified groups over a local non-archimedean field. We prove that the corresponding torus weight spaces are supported in exactly one cohomological degree, and are pairwisely non-isomorphic irreducible representations of the pro-unipotent radical of the corresponding parahoric subgroup. We also prove that all Moy--Prasad quotients of this stratum are maximal varieties, and we investigate the relation between the resulting representations and Kirillov's orbit method.
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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 representations.