Decomposition of higher Deligne-Lusztig representations

Higher Deligne-Lusztig representations are virtual smooth representations of parahoric subgroups in a $p$-adic group. They are natural analogs of classical Deligne-Lusztig representations of reductive groups over finite fields. The most interesting higher Deligne-Lusztig representations are those attached to elliptic maximal tori, whose compact inductions are expected to realize supercuspidal representations of $p$-adic groups. Under a mild condition on $p$, in this paper we establish an explicit decomposition of these higher Deligne-Lusztig representations into irreducible summands. Surprisingly, all the irreducible summands are built in the same way as those in Yu's construction of irreducible supercuspidal representations, the only difference being that the Weil-Heisenberg representations in Yu's construction are replaced by their geometric analogs. As an application, we show that each irreducible supercuspidal representation of a $p$-adic group, attached to an unramified cuspidal datum, is a direct summand of the compact induction of a suitable higher Deligne-Lusztig representation, whenever the cardinality of the residue field of the $p$-adic field is not too small.