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All supercuspidal representations of \(SL_{\ell}\) over a p-adic field are induced. (English) Zbl 0538.22011

Representation theory of reductive groups, Proc. Conf., Park City/Utah 1982, Prog. Math. 40, 185-196 (1983).
[For the entire collection see Zbl 0516.00013.]
Let F be a finite extension of \({\mathbb{Q}}_ p\), \({\mathcal O}\) the ring of integers, \(\omega\) a uniformizing element. Let \(\ell\) be a prime and put \(G=GL(\ell,F)\), \(\bar G=SL(\ell,F)\), \(\bar K=SL(\ell,{\mathcal O})\). The authors prove (Theorem 3.1) that any irreducible supercuspidal representation of \(\bar G\) is induced from either \(\bar B\), the Iwahori subgroup of \(\bar K\), or one of the subgroups \(\bar K^ w\), where \(w=diag(\omega^ r,1,...,1) (r=0,1,...,\ell -1).\) The proof uses Mackey’s theorem and the classification of supercuspidal representations of G due to H. Carayol.
Reviewer: A.V.Zelevinsky

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11S37 Langlands-Weil conjectures, nonabelian class field theory

Citations:

Zbl 0516.00013