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Formal degrees and adjoint \(\gamma\)-factors. (English) Zbl 1131.22014

Let \(F\) be a local field and let \(G\) be a reductive algebraic group defined over \(F\). Let \(\pi\) be a discrete series representation of \(G(F)\). The adjoint \(\gamma\)-factor \(\gamma(s,\pi,r,\psi)\) (\(r\) a finite dimensional representation of the \(L\)-group of \(G\) and \(\psi\) is an additive character of \(F\)) is defined in terms of \(L\)-functions and \(\varepsilon\)-factors. The authors of this paper posit a relationship between the formal degree of the representation, and the absolute value of the adjoint \(\gamma\)-factor. They prove the formula in a large number of special cases. The proofs make use of much delicate information about the groups in question and their discrete series representations.

MSC:

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