## 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.

### MSC:

 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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### References:

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