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 ${\kappa }^{\mathrm{\prime }}$ of the Weil-Heisenberg representation $\kappa$. In this note, we show that ${\kappa }^{\mathrm{\prime }}$ and $\kappa$ differs by a character $\chi$. Moreover, under a mild condition on the cardinality $q$ of the residue field (for instance $q>3$), we show that $\chi$ 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.