On holomorphic theta functions associated to rank r isotropic discrete subgroups of a g-dimensional complex space

2015/06/21 by Allal Ghanmi, Ghanmi, Allal, Ahmed Intissar +3
Mathematics · #14K25 #33E05 #Advanced Algebra and Geometry #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Primary 32N05 #Secondary 37J05 #math.CV #msc:14K25 #msc:32N05 #msc:33E05 #msc:37J05

paper · pdf · doi:10.48550/arxiv.1506.06353

13 pages

arxiv created 2015/06/21 · openalex publication_date 2015/06/21 · arxiv updated 2015/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are interested in the L²-holomorphic automorphic functions on a g-dimensional complex space V^g_ℂ endowed with a positive definite hermitian form and associated to isotropic discrete subgroups Γ of rank 2≤ r ≤ g. We show that they form an infinite reproducing kernel Hilbert space which looks like a tensor product of a theta Fock-Bargmann space on V^r_ℂ=Span_ℂ(Γ) and the classical Fock-Bargmann space on V^g-r_ℂ. Moreover, we provide an explicit orthonormal basis using Fourier series and we give the expression of its reproducing kernel function in terms of Riemann theta function of several variables with special characteristics.

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