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.
Chan,The cohomology of semi-infinite Deligne-Lusztig varieties, J
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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.