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Local integrability of character distributions of \(\text{GL}_ N (F)\) where \(F\) is a local nonarchimedean field of arbitrary characteristic. (Intégrabilité locale des caractères-distributions de \(\text{GL}_ N (F)\) où \(F\) est un corps local non-archimédien de caractéristique quelconque.) (French) Zbl 0856.22024
Let \(F\) be a non-Archimedean local field. The character of an irreducible admissible representation of \(G= GL(n, F)\) is a locally integrable function. If \(\text{char}(F)= 0\), this is a special case of a well-known result of Harish-Chandra. In the present paper, the theorem is proved for any characteristic. One has to study the distribution in the neighbourhood of any semisimple element of \(G\). This has been done by Rodier for separable semisimple elements [F. Rodier, Duke Math. J. 52, 771-792 (1985; Zbl 0609.22004)]. In the present paper, the problem is solved for any semisimple element \(y\) by a refinement of Rodier’s analysis, using an idea of Bushnell and Kutzko, which enables one to reduce the problem to the study of an invariant distribution on the Lie algebra of the centralizer of \(y\) in \(G\).

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
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