## 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

Zbl 0516.00013