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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:

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