  The Bruhat-Tits theory is a key ingredient in the construction of irreducible
smooth representations of $p$-adic reductive groups. We describe
generalizations to arbitrary such representations of several results recently
obtained in the case of supercuspidal representations, in particular regarding
the local Langlands correspondence and the internal structure of the
$L$-packets.
  We prove that the enhanced $L$-parameters with semisimple cuspidal support
are those which are obtained via the (ordinary) Springer correspondence.
  Let ${\mathbf G}$ be a connected reductive group over a non-archimedean field
$F$ of residual characteristic $p$. In the case where ${\mathbf G}$ splits over
a tamely ramified extension of $F$ and $p$ does not divide the order of the
Weyl group of ${\mathbf G}$, we show that the enhanced $L$-parameters with
semisimple cuspidal support correspond to the irreducible smooth
representations of ${\mathbf G}(F)$ with non-singular supercuspidal support via
the local Langlands correspondence constructed by Kaletha, under the assumption
that the latter satisfies certain expected properties. As a consequence, we
obtain that every compound $L$-packet of ${\mathbf G}(F)$ contains at least one
representation with non-singular supercuspidal support.
