Types for tame \(p\)-adic groups. (English) Zbl 1492.22013

Let \(k\) stand for a non-Archimedean local field of residual characteristic \(p\), and let us denote by \(G\) a connected reductive group over \(k\), which splits over a tamely ramified extension of \(k\). Let us assume that \(p\) does not divide the order of the Weyl group of \(G.\)
In the paper under the review, the author attaches to an irreducible smooth representation of \(G(k)\) a so-called datum, which is a tuple of the form \[(x, (X_i)_{1 \leq i \leq n}, (\rho_0, V_{\rho_0})),\] where \(n\) is an integer, \(x\) belongs to the Bruhat-Tits building of \(G\), \(X_i \in \mathfrak{g}^{\ast}\), for \(1 \leq i \leq n\), satisfying certain conditions, while \((\rho_0, V_{\rho_0})\) is an irreducible representation of a finite group, obtained as the reductive quotient of the special fiber of the connected parahoric group scheme attached to the derived group of a twisted Levi subgroup of \(G\).
Notion of the datum presents a refinement of the unrefined minimal K-type, introduced by A. Moy and G. Prasad [Invent. Math. 116, No. 1–3, 393–408 (1994; Zbl 0804.22008); Comment. Math. Helv. 71, No. 1, 98–121 (1996; Zbl 0860.22006)]. First main result in the paper is the existence of the maximal datum for any irreducible representation of \(G(k)\), which enables one to produce the input required for the construction of types as in the work of J.-L. Kim and J.-K. Yu [Prog. Math. 323, 337–357 (2017; Zbl 1409.22012)].
This enables the author to prove the second main results of the paper, which states that every smooth irreducible representation of \(G(k)\) contains one of the types constructed by Kim-Yu. Finally, in the third main result, the author uses elements \(X_i\) belonging to the datum to provide appropriate characters of the twisted Levi subgroups and representations \((\rho_0, V_{\rho_0})\) to obtain a depth-zero supercuspidal representations, using the study of Weil representations, which leads to the fact that all irreducible supercuspidal representations of \(G(k)\) can be obtained using the Yu’s construction inducing from open, compact mod center subgroups of \(G(k)\) [J.-K. Yu, J. Am. Math. Soc. 14, No. 3, 579–622 (2001; Zbl 0971.22012)].
We note that the last description of supercuspidal has also been obtained by J.-L. Kim [J. Am. Math. Soc. 20, No. 2, 273–320 (2007; Zbl 1111.22015)], for non-Archimedean local fields of the characteristic zero and under the assumption that \(p\) is very large. On the other hand, proof given in the paper under the review works for non-Archimedean local fields of arbitrary characteristic, with the only requirement that \(p\) does not divide the order of the Weyl group.


22E50 Representations of Lie and linear algebraic groups over local fields
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