\(\varepsilon\) factor of representations of classical groups. (English) Zbl 0599.12012

Piatetski-Shapiro, Ilya I.; Rallis, Stephen

Proc. Natl. Acad. Sci. USA 83, No. 13, 4589-4593 (1986).

The authors present a theory of local and global zeta integrals (and \(L\)-functions) associated with the group of isometries of a non-degenerate symmetric or skew-symmetric form on a vector space over a local or global field and settle the subtle question of determining the correct ”\(\varepsilon\)-factors” for \(L\)-functions attached to (relevant) local representations. Their results extend the work of R. Godement and H. Jacquet [Zeta functions of simple algebras. Lecture Notes in Mathematics. 260. Berlin etc.: Springer (1972; Zbl 0244.12011)] in the case of \(\text{GL}_n\) but need special intertwining operators with ‘proper normalization’ (in contrast with the use of the Fourier transform in the case of \(\text{GL}_n\).

Reviewer: S. Raghavan

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
22E55	Representations of Lie and linear algebraic groups over global fields and adèle rings
11S45	Algebras and orders, and their zeta functions

Keywords:

classical groups; automorphic forms; theory of local and global zeta integrals; L-functions; \(\varepsilon\)-factors; intertwining operators