## 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

Zbl 0685.22009
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### References:

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