Achab, Dehbia Representations of rank 2 Jordan algebras and associated zeta functions. (Représentations des algèbres de Jordan de rang 2 et fonctions zêta associées.) (French) Zbl 0824.11054 Ann. Inst. Fourier 45, No. 2, 437-451 (1995). Summary: In a previous work [C. R. Acad. Sci., Paris, Sér. I 316, 977-982 (1993; Zbl 0780.11038)] the author defined the zeta function attached to a Jordan algebra representation and a lattice in the representation space and established that it converges on a half plane, admits a meromorphic continuation and satisfies a functional equation. This function is a generalization of the Koecher zeta function; it is given, on its convergence domain, by a series, which sums over some of the lattice elements, modulo some arithmetic subgroup \(\Gamma_ 0\) which fixes the lattices. The present article is about the explicit study of this zeta function in the particular case of a rank 2 Jordan algebra \(V\) considering some representation in a Clifford algebra in which the lattice is chosen. Therefore, the group \(\Gamma_ 0\) is equal to some arithmetic subgroup of some covering of the orthogonal group \(SO_ 0 (1,n)\) where the dimension of \(V\) is set to be \(n+1\). In this case, the zeta function is new, and it satisfies some functional equation very similar to the one of the Riemann zeta-function, the Euler gamma function being replaced by the gamma function of the Lorentz cone. Cited in 1 Document MSC: 11M41 Other Dirichlet series and zeta functions 17C99 Jordan algebras (algebras, triples and pairs) 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) 14M17 Homogeneous spaces and generalizations Keywords:Jordan algebras; Clifford algebras; arithmetic group; zeta functions; zeta function attached to a Jordan algebra representation; rank 2 Jordan algebra; functional equation Citations:Zbl 0780.11038 PDFBibTeX XMLCite \textit{D. Achab}, Ann. Inst. Fourier 45, No. 2, 437--451 (1995; Zbl 0824.11054) Full Text: DOI Numdam EuDML References: [1] [1] , Fonction zêta des représentations des algèbres de Jordan, thèse Université Paris 6, 1993. · Zbl 0780.11038 [2] [2] , Fonction zêta d’une représentation d’algèbre de Jordan, C.R.A.S, Paris, t. 316, Série I (1993), 977-982. · Zbl 0780.11038 [3] [3] , Représentations d’une algèbre de Jordan, polynômes invariants et harmoniques de Stiefel, J. reine angew. Math., 423 (1992), 47-71. · Zbl 0749.17042 [4] [4] et , Analysis on symmetric cones (preprint). · Zbl 0841.43002 [5] [5] et , Generalized Bessel Functions, Journal of Funct. Analysis, 71, No. 1 (1987). · Zbl 0612.43009 [6] [6] , Uber Dirichlet-Reihen mit Funktionalgleichung, J. reine angew. Math., 192 (1953), 1-23. · Zbl 0053.05501 [7] [7] , Kcher-Maass- Series for Modular Forms of Quaternions, Manuscripta Math., 66 (1990), 431-451. · Zbl 0698.10019 [8] [8] , Série discrète pour les groupes de Lorentz SOo(2n, 1), Colloque sur les fonctions sphériques et la théorie des groupes, Nancy, 1971. · Zbl 0335.22012 [9] [9] , Mobius Transformations in Several Dimensions, Adv. Math., 101 (1993), 87-113. · Zbl 0793.15019 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.