## The admissible dual of $$SL(N)$$. I.(English)Zbl 0787.22017

For a non-Archimedean local field $$F$$ and a reductive group $$G$$ over $$F$$ the determination of the supercuspidal representations of $$G$$ is a delicate matter. Consider the groups $$G=GL_ n(F)$$ and $$G'= SL_ n(F)$$. Any irreducible smooth $$G$$-representation restricts to a multiplicity free finite direct sum of smooth irreducible $$G'$$-representations and any two $$G$$-representations having in their restrictions a $$G'$$- representation in common, differ by a character of the determinant. Furthermore supercuspidal representations restrict to supercuspidals.
Using these properties of the restriction and their theory of simple types of $$GL_ n(F)$$ the authors are able to prove that any supercuspidal representation of $$SL_ n(F)$$ is induced from the norm-1 group of a principal order in the algebra $$\text{Mat}_ n(F)$$. Specializing to tamely ramified representations (=representations generated by Iwahori fixed vectors) they show that tamely ramified representations have tamely ramified subquotients.

### MSC:

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