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The Bruhat-Tits theory is a key ingredient in the construction of irreduciblesmooth representations of $p$-adic reductive groups. We describegeneralizations to arbitrary such representations of several results recentlyobtained in the case of supercuspidal representations, in particular regardingthe local Langlands correspondence and the internal structure of the$L$-packets.  We prove that the enhanced $L$-parameters with semisimple cuspidal supportare 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 overa tamely ramified extension of $F$ and $p$ does not divide the order of theWeyl group of ${\mathbf G}$, we show that the enhanced $L$-parameters withsemisimple cuspidal support correspond to the irreducible smoothrepresentations of ${\mathbf G}(F)$ with non-singular supercuspidal support viathe local Langlands correspondence constructed by Kaletha, under the assumptionthat the latter satisfies certain expected properties. As a consequence, weobtain that every compound $L$-packet of ${\mathbf G}(F)$ contains at least onerepresentation with non-singular supercuspidal support.

Bruhat-Tits buildings, representations of $p$-adic groups and Langlands correspondence