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On the local constancy of characters. (English) Zbl 1071.22023

Let \(F\) be a non-Archimedean local field of characteristic zero and let \(G\) be a connected reductive algebraic group defined over \(F\). Let \(\pi\) be an irreducible admissible representation of \(G(F)\). Then Harish-Chandra defined the character \(\Theta_\pi\) of \(\pi\) as a distribution and showed that it is represented by a locally integrable function which is locally constant on the regular elements of \(G(F)\). In this paper the author develops an alternative approach to these questions, due to R. Howe, to make explicit what “locally constant” means, i.e. for any point of \(G(F)\) he gives a neighbourhood of that point on which \(\Theta_\pi\) is constant.

MSC:

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