# Twisted Levi sequences in GLN and generic elements associated to minimal elements

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## Twisted Levi sequences in GLN and generic elements associated to minimal elements

In this section, we give an example of tamely ramied twisted Levi sequence and an example of generic element. This generic element comes from a minimal element relatively to a nite tamely ramied eld extension. More precisely, let E0=E=F be a tower of tamely ramied eld extensions and let V be an E0-vector space of dimension d. We are going to dene and describe explicitly the groups scheme H0 = ResE0=F AutE0(V ), H = ResE=F AutE(V ) and G = AutF (V ). We will show that the sequence (H0;H;G) forms a tamely ramied twisted Levi sequence in G. The choice of an E0-maximal decomposition D, V = (V1: : :Vd), of V in 1-dimensional E0-vector spaces gives birth to a maximal torus TD of AutE0(V ). By restriction of scalar, we get a maximal torus T = ResE0=E(TD) of H0. We are going to describe the set over F of roots of H0 and H with respect to T. Moreover we will describe the condition GE1 in this situation. Finally, given c 2 E0 minimal over E, we will introduce an element X sr(c) 2 Lie(Z(H0)) and prove that it satises GE1 and is H-generic.

### Abstract twisted Levi sequences

In this subsection, we prove algebraic facts that will be applied to the following subsections. We start with a very easy and well-known lemma. Let f be commutative ring and B be a commutative f-algebra, C be an B-algebra. Let A be an f-algebra. In this situation A f B is an B-algebra and C is naturally an f-algebra. Lemma 1.6.2. With the previous notations, the C-algebra (A f B) B C is canonically isomorphic to A f C. Explicitly, the isomorphism is given by (A f B) B C ! A f C (a b) c 7! a bc.

#### Factorization of tame simple characters

Let [A; n; r; ] be a tame simple stratum. In this section, we choose and x a dening sequence f[A; n; ri; i], 0 i sg and a simple character 2 C(A; 0; ) , we show that = Ys i=0 i where i satises some conditions. We then introduce two cases depending on the condition that s 2 F or s 62 F.

Abstract factorizations of tame simple characters

Fix a tame simple stratum [A; n; r; ] in the algebra A = EndF (V ). Propositions 1.4.3 and 1.4.4 allow us to choose a dening sequence f[A; n; ri; i], 0 i sg (see corollary 1.2.11) such that, putting Bi := A \ EndF[i](V ) and r0 = 0, 0 = the following holds.
(vii) F[i+1] ( F[i] for 0 i s 􀀀 1 (vi’) The stratum [Bi+1; ri+1; ri+1 􀀀 1; i 􀀀 i+1] is simple in the algebra EndF[i+1](V ) for 0 i s 􀀀 1. We x such a dening sequence in the rest of this section 1.7, this includes the following subsection 1.7.2.
The elements i , 0 i s are all included in F[]. Put Ei := F[i] for 0 i s. Let us dene elements ci , 0 i s , thanks to the following formulas. ci = i 􀀀 i+1 if 0 i s 􀀀 1 cs = s The following proposition is the factorisation of tame simple characters.

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Generic characters associated to tame simple characters

We continue with the same notations as in section 1.7. Thus we have a xed tame simple stratum [A; n; 0; ] and various objects and notations relative to it. In particular we have a dening sequence and a simple character 2 C(A; 0; : We have also distinguished two cases. In both (Case A) and (Case B), we have introduced various objects and notations and have established results relative to them. In this section we are going to introduce a 4-uple ( 􀀀!G ; y;􀀀!r ; 􀀀! which will be part of a complete Yu datum.

Introduction
1 Comparison of constructions of supercuspidal representations: from Bushnell-Kutzko’s construction to Yu’s construction
1.1 Intertwining, compact induction and supercuspidal representations
1.2 Bushnell-Kutzko’s construction of supercuspidal representations for GLN
1.2.1 Simple strata
1.2.2 Simple characters
1.2.3 Simple types and representations
1.3 Yu’s construction of tame supercuspidal representations
1.3.1 Tamely ramied twisted Levi sequences and groups
1.3.2 Generic elements and generic characters
1.3.3 Yu data
1.3.4 Yu’s construction
1.4 Tame simple strata
1.5 Minimal elements and standard representatives
1.6 Twisted Levi sequences in GLN and generic elements associated to minimal elements
1.6.1 The group schemes of automorphisms of a free A- module of nite rank
1.6.2 Trace of endomorphisms and base change
1.6.3 Abstract twisted Levi sequences
1.6.4 Tame twisted Levi sequences
1.6.5 Generic elements associated to minimal elements .
1.7 Factorization of tame simple characters
1.7.1 Abstract factorizations of tame simple characters
1.7.2 Explicit factorizations of tame simple characters
1.8 Generic characters associated to tame simple characters
1.8.1 The characters i associated to a factorization of a tame simple character
1.8.2 The characters ^i
1.9 Extensions and main theorem of the comparison: from Bushnell- Kutzko’s construction to Yu’s construction
2 Analytic ltrations
2.1 Schemes
2.1.1 Generalities
2.1.2 Higher dilatations and congruence subgroups
2.2 Berkovich k-analytic spaces
2.2.1 k-anoid algebras
2.2.2 k-anoid spaces
2.2.3 k-analytic spaces
2.3 Bruhat-Tits buildings and Moy-Prasad ltrations
2.4 Denitions and rst properties of analytic ltrations
2.4.1 Notions of potentially Demazure objects
2.4.2 Filtrations of rational potentially Demazure k-anoid groups
2.4.3 Filtrations of Lie algebra
2.5 Filtrations associated to points in the Bruhat-Tits building .
2.5.1 Denitions and properties of Gx;r and
2.5.2 A cone
2.5.3 Comparison with Moy-Prasad ltrations in the tame case
2.5.4 Filtrations of the Lie algebra