  A higher Deligne-Lusztig representation is a virtual smooth representation of
a parahoric subgroup in a $p$-adic group, which is constructed from the
Deligne-Lusztig induction. In this note, under a mild condition on $p$ we
provide a decomposition of higher Deligne-Lusztig representations, attached to
an unramified elliptic maximal torus, into irreducible representations obtained
in a similar way as those used in Yu's construction of supercuspidal
representations of $p$-adic groups. As a consequence, we show that all the
unramified supercuspidal representations are direct summands of inductions of
higher Deligne-Lusztig representations.
