## Types and Hecke algebras for principal series representations of split reductive $$p$$-adic groups.(English)Zbl 0903.22009

In [The admissible dual of $$GL(N)$$ via compact open subgroups. Annals of Mathematics Studies. 129 (Princeton 1993; Zbl 0787.22016)] C. Bushnell and P. Kutzko described the admissible dual of $$G=GL(F)$$ for a $$p$$-adic field $$F$$ in terms of “types”, that is, in terms of certain finite dimensional representations of certain compact open subgroups. If a representation $$\pi$$ contains a type it can by Frobenius reciprocity be found in the representation induced from that type. This is the way that in loc. cit. all representations of $$G$$ are constructed. The irreducible representations containing a given type are given by the simple modules of the corresponding Hecke algebra so that the classification proceeds in terms of Hecke modules then. In the paper under consideration types in the above sense are constructed and given explicitly for $$G$$ being any split reductive group and $$\pi$$ being a principal series representation. The corresponding Hecke algebras are described explicitly. In addition, it is described how parabolic induction and Jaquet restriction and questions about square integrability can be transferred to this context.

### MSC:

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

Zbl 0787.22016
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