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Character expansions and unrefined minimal K-types. (English) Zbl 1037.22035

Let \(k\) be a \(p\)-adic field of characteristic zero and G be a connected reductive group defined over \(k\). Let \(G\) be the group of \(k\)-rational points of G and \(g\) be the Lie algebra of \(G\). Let us define \(E(G)\) as the set of equivalence classes of irreducible admissible representations of \(G\). For \((\pi, V_\pi) \in E(G)\), let \(\Theta_\pi\) be the character of \(\pi\). It is demonstrated that if an irreducible admissible representation \((\pi, V_\pi)\) of positive depth \(\rho\) contains a good type, then \(\Theta_\pi\) has a character expansion, called a \(\Gamma\)-asymptotic expansion which is valid on the \(G\)-domain \(g_\rho\). The latter appears to be larger compared to the case where the Harish-Chandra-Howe character expansion is known to be valid.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
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