Hecke algebra isomorphisms for GL(n) over a p-adic field
The results on the existence and uniqueness of minimal K-types for irreducible admissible representations of \(G=GL(n,F)\) (F a p-adic field) that have recently been obtained by the same authors [Astérisque 171- 172, 257-273 (1989; see the preceding review Zbl 0715.22018)] are refined and then used to obtain further progress towards the classification of the irreducible admissible representations of G. In particular for the tame case it is shown that to each irreducible admissible representation \(\pi\) of G that is square integrable modulo the center of G one can associate a parahoric subgroup K with a representation \(\theta\) of K containing a minimal K-type of \(\pi\), an extension E of F and a subgroup \(G'=GL(m,E)\subset G\) with \(m(E:F)=n\) such that \(K\cap G'=B'\) is an Iwahori subgroup of \(G'\) and such that one has a Hecke algebra isomorphism preserving supports between \(H(G'//B',1)\) and H(G//K,\(\theta\)). Furthermore the pair (K,\(\theta\)) has multiplicity one in \(\pi\) in analogy with the minimal K-types in the real case.
scientific article; zbMATH DE number 4177416
scientific article