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Unrefined minimal \(K\)-types for \(p\)-adic groups. (English) Zbl 0804.22008

Let \(G\) be a reductive group over a \(p\)-adic field of characteristic zero. It is widely believed that a classification of the admissible dual of \(G\) may be achieved via studying the restriction of an admissible representation of \(G\) to certain compact open subgroups. To this end the first author gave [in Contemp. Math. 86, 249-254 (1989; Zbl 0685.22009)] the definition of a minimal \(K\)-type as a certain type of representation of parahoric filtration subgroups. Contrary to the real case the definition of a minimal \(K\)-type is intrinsic in that it does not involve a given admissible representation of \(G\). To use it for classification arguments one thus must prove that every admissible representation of \(G\) contains a minimal \(K\)-type. In the paper under consideration this is established.
The author uses an isomorphism of the graded parts of the parahoric filtration to the graded parts of a corresponding Lie algebra filtration which allows him to switch over to the Lie algebra. Via the minimal \(K\)- types one now can attach to a given admissible representation \(\pi\) a rational number \(\rho(\pi)\), the depth of the minimal \(K\)-type, which should play a key rôle in the classification of the admissible dual similar to the real case.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
17B70 Graded Lie (super)algebras
20G25 Linear algebraic groups over local fields and their integers
22E60 Lie algebras of Lie groups

Citations:

Zbl 0685.22009
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References:

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