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**Local tame lifting for \(GL(N)\). I: Simple characters.**
*(English)*
Zbl 0878.11042

Let \(K/F\) be a finite separable extension of nonarchimedean local fields and \(N\) a natural number. The base change conjecture of Langlands demands the existence of a process \(\pi\mapsto l_{K/F}(\pi)\) attaching to an irreducible smooth representation \(\pi\) of \(GL_N(F)\) a representation \(l_{K/F}(\pi)\) of the same sort of the group \(GL_N(K)\). As the authors show, the base change map \(l_{K/F}\) can be constructed for \(\text{char}(F)=0\) and \(K/F\) tamely ramified, we assume this from now on. The aim of the present paper is to give a first step towards presenting the base change in explicit local terms. The authors refer to the classification of the irreducible smooth representations of \(GL_N(F)\) by C. Bushnell and P. Kutzko [in: The admissible dual of \(GL(N)\) via compact open subgroups, Ann. Math. Stud. 129, Princeton University Press, Princeton (1993; Zbl 0787.22016)]. In this book it is shown that every supercuspidal representation contains a simple type. A simple type is a pair \((K,\tau)\) consisting of a compact open subgroup \(K\) of \(GL_N(F)\) and an irreducible representation \(\tau\) of \(K\), both of a very special kind. An admissible representation \(\pi\) of \(GL_N(F)\) is then said to contain the simple type \((K,\tau)\) if its restriction \(\pi|_K\) contains \(\tau\). In that case the \(K\)-isotype of \(\tau\) in \(\pi\) forms a finite-dimensional module under the corresponding Hecke algebra which is irreducible if and only if \(\pi\) is. It is shown that the corresponding Hecke algebra is naturally isomorphic to the Iwahori-Hecke algebra, which reduces the classification problem to the representations with Iwahori-fixed vector. The latter are classified by D. Kazhdan and G. Lusztig in their proof of the Deligne-Langlands conjecture [Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)].

One main ingredient of the construction of simple types in Bushnell-Kutzko (loc. cit.) is a simple character, which is a special abelian character of some compact open subgroup. In the present paper the authors give a lifting construction of simple characters \(\theta_F\) of \(GL_N(F)\) to simple characters \(\theta_K\) of \(GL_N(K)\). The main theorem then is that an irreducible smooth representation \(\pi\) of \(GL_N(F)\) contains the simple character \(\theta_F\) if and only if its base change \(l_{K/F}(\pi)\) contains \(\theta_K\). To give a full explicit description of base change, it remains to extend this to a lift of simple types and then define an analogous lifting for split types, i.e. the counterpart to simple types, constructed by Bushnell and Kutzko to take care of those representations that do not contain a simple type.

The basic idea of the lifting construction is this: let \(F[\beta]/F\) be a finite extension. The algebra \(K\otimes_F F[\beta]\) will be a product of fields \(E_i=F[\beta_i]\). The attachment \(\beta\mapsto (\beta_i)_i\) then gives the basic scheme of the lifting process. The authors check that their lifting preserves conjugation and induction relations of simple characters as is to be required.

Although base change is only available in characteristic zero the constructions of this paper are valid in any characteristic. So, if some unsurprising facts together with base change become available in positive characteristic the results of this paper also will.

One main ingredient of the construction of simple types in Bushnell-Kutzko (loc. cit.) is a simple character, which is a special abelian character of some compact open subgroup. In the present paper the authors give a lifting construction of simple characters \(\theta_F\) of \(GL_N(F)\) to simple characters \(\theta_K\) of \(GL_N(K)\). The main theorem then is that an irreducible smooth representation \(\pi\) of \(GL_N(F)\) contains the simple character \(\theta_F\) if and only if its base change \(l_{K/F}(\pi)\) contains \(\theta_K\). To give a full explicit description of base change, it remains to extend this to a lift of simple types and then define an analogous lifting for split types, i.e. the counterpart to simple types, constructed by Bushnell and Kutzko to take care of those representations that do not contain a simple type.

The basic idea of the lifting construction is this: let \(F[\beta]/F\) be a finite extension. The algebra \(K\otimes_F F[\beta]\) will be a product of fields \(E_i=F[\beta_i]\). The attachment \(\beta\mapsto (\beta_i)_i\) then gives the basic scheme of the lifting process. The authors check that their lifting preserves conjugation and induction relations of simple characters as is to be required.

Although base change is only available in characteristic zero the constructions of this paper are valid in any characteristic. So, if some unsurprising facts together with base change become available in positive characteristic the results of this paper also will.

Reviewer: A.Deitmar (Heidelberg)

### MSC:

11S37 | Langlands-Weil conjectures, nonabelian class field theory |

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

### Keywords:

tamely ramified extensions; local tame lifting; simple type; supercuspidal representations; general linear group; irreducible representations; Hecke algebra; Iwahori-fixed vector; base change conjecture; base change map; lifting construction of simple characters
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\textit{C. J. Bushnell} and \textit{G. Henniart}, Publ. Math., Inst. Hautes Étud. Sci. 83, 105--233 (1996; Zbl 0878.11042)

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