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On characters of irreducible unitary representations of general linear groups. (English) Zbl 0856.22026
The author reduces the determination of the characters of irreducible unitary representations of \(\text{GL}(n)\) over a non-Archimedean local field to the determination of the characters of the irreducible square integrable representations of \(\text{GL}(m)\) with \(m\leq n\). He also obtains a formula expressing the characters of irreducible unitary representations in terms of the characters of the segment representations of Zelevinsky. This formula generalizes the classical formula for the Steinberg character of \(\text{GL}(n)\). To obtain his results, he uses a clever trick based on the formal similarity of the unitary duals in the Archimedean and the non-Archimedean case and a result of Gregg Zuckerman expressing the character of the trivial representation as a linear combination of standard characters for \(\text{GL}(n)\) over complex numbers.

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