An explicit decomposition of higher Deligne-Lsuztig representations

In a previous paper, the second named author obtains a decomposition of an elliptic higher Deligne-Lusztig representation into irreducible summands, which are built in the same way as Yu types using a geometric analog κ' of the Weil-Heisenberg representation κ. In this note, we show that κ' and κ differs by a character χ. Moreover, under a mild condition on the cardinality q of the residue field (for instance q > 3), we show that χ equals the quadratic character constructed by Fintzen-Kaletha-Spice, which gives an explicit irreducible decomposition result on elliptic higher Deligne-Lusztig representations. As an application, we deduce (under the mild condition on q) that each unramified Yu type appears in the cohomology of higher Deligne-Lusztig varieties, and each unramified Kaletha's regular supercuspidal representation is the compact induction of a specified higher Deligne-Lusztig representation up to a sign.