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Induced representations of GL(n,A) for p-adic division algebras A. (English) Zbl 0684.22008
Several results on the representation theory of GL(n,A) over a p-adic division algebra A are proved. They generalize the corresponding results obtained by A. V. Zelevinsky for GL(n) over a p-adic field [in Ann. Sci. Ec. Norm. Supér., IV. Sér. 13, 165-210 (1980; Zbl 0441.22014)]. The main motivation for considering the mentioned results is the unitarizability problem for GL(n,A). Consequences for the unitarizability are discussed at the end of the paper. Also a consequence for GL(n) over a field is observed in the paper. The paper continues the investigation of the representation theory of GL(n,A) started by P. Deligne, D. Kazhdan and M.-F. Vignéras [in Représentations des groupes réductifs sur un corps local, 33-117 (1984; Zbl 0583.22009)].
 22E50 Representations of Lie and linear algebraic groups over local fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F33 Congruences for modular and $$p$$-adic modular forms 16Kxx Division rings and semisimple Artin rings 11S45 Algebras and orders, and their zeta functions