Functional equations of prehomogeneous zeta functions and intertwining operators. (English) Zbl 1161.11413

Summary: We establish a relation between the gamma matrices of the functional equations satisfied by zeta functions associated with prehomogeneous vector spaces and certain integrals related to the intertwining operator of degenerate principal series representations of general linear groups.


11S90 Prehomogeneous vector spaces
11M41 Other Dirichlet series and zeta functions
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
Full Text: DOI Euclid


[1] R. Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math., 260 , Springer-Verlag, 1972. · Zbl 0244.12011
[2] I. M. Gelfand and G. I. Shilov, Generalized functions, I, Academic Press, New York, 1964.
[3] M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima and M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. Math., 107 (1978), 1-39. · Zbl 0377.43012
[4] T. Oshima, Generalized Capelli identities and boundary value problems for \(GL(n)\), In: Structures of solutions of differential equations, Katata/Kyoto 1995, (eds. M. Morimoto and T. Kawai), World Scientific, 1996, pp.,307-335. · Zbl 0909.22024
[5] S. Gelbart, I. Piatetski-Shapiro and S. Rallis, Explicit constructions of automorphic \(L\)-functions, Lecture Notes in Math., 1254 , Springer-Verlag, 1987. · Zbl 0612.10022
[6] F. Sato, Zeta functions in several variables associated with prehomogeneous vector spaces I: Functional equations, Tôhoku Math. J., 34 (1982), 437-483. · Zbl 0497.14007
[7] F. Sato, Eisenstein series on weakly spherical homogeneous spaces and zeta functions of prehomogeneous vector spaces, Comment. Math. Univ. St. Pauli, 44 (1995), 129-150. · Zbl 0840.11035
[8] F. Sato, Eisenstein series on weakly spherical homogeneous spaces of \(GL(n)\), Tôhoku Math. J., 50 (1998), 23-69. · Zbl 0944.11017
[9] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their invariants, Nagoya Math. J., 65 (1977), 1-155. · Zbl 0321.14030
[10] M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math., 100 (1974), 131-170. · Zbl 0309.10014
[11] T. Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan, 24 (1972), 132-188. · Zbl 0227.10031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.