Bruhat-Tits buildings, representations of \(p\)-adic groups and Langlands correspondence (Q6568810)

Based on the theory of Bruhat-Tits buildings, the author analyzed the local Langlands correspondence (LLC) and the Springer correspondence to study the supercuspidal representations of \(p\)-adic reductive groups and the structure of the corresnponding L-packets.\N\NLet \(G\) be a connected reductive group, that is split over a tamely ramified extension of a non-Archimedean field \(F\) of residual characteristic \(p\) not dividing the order of the Weyl group of \(G\). The group \(G\) is acting on its Bruhat-Tits building \(\mathcal B(G,F)\) providing the parahoric subgroups \(G_{x,0}\) and the Moy-Prasad filtration \(G_{x,0}\triangleright G_{x,r_1} \triangleright G_{x,r_1} \triangleright G_{x,r_1} \dots\), where \(0 <r_1 <r_2< \dots\). \NAdler constructed all supercuspidal representations of \(G\).Yu generalized the Adler's construction of all supercuspidal representations of \(G\). A supercuspidal representation is non-singular if the corresponding discrete \(L\)-parameters have a trivial restriction to \(\mathrm{SL}_2(\mathbb C)\). The author generalizes this notion for an arbitrary irreducible smooth representation by replacing that condition by a condition that its \textit{support is regular} (or \textit{non-singular}) (Definition 4.3.7). By analyzing the Springer correspondence and the LLC, the author obtains the main result (Theorem 4.6.3) stating that the enhanced L-parameters with semisimple cuspidal support correspond to irreducible smooth representations of \(G(F)\) with non-singular supercuspidal support via the Kaletha's simplification of Yu's construction of LLC under certain conditions, namely the parameters with non-singular supercuspidal support. The author also deduces that every compound L-packet of \(G(F)\) contains at least one representation with non-singular supercuspidal support.