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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

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.

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

Citations:

Zbl 0780.11038
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References:

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