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Journal of Lie Theory 18 (2008), No. 4, 851--867
Copyright Heldermann Verlag 2008

A Satake Type Theorem for Super Automorphic Forms

Roland Knevel
Université de Luxembourg, Unité de Recherche en Mathématiques, Av. de la Faïencérie 162a, 1511 Luxembourg


The aim of this article is a Satake type theorem for super automorphic forms on a complex bounded symmetric super domain $\cal B$ of rank $1$ with respect to a lattice $\Gamma$. 'Super' means: additional odd (anticommuting) coordinates on an ordinary complex bounded symmetric domain $B$ (the so-called body of $\cal B$) of rank $1$. Satake's theorem says that for large weight $k$ all spaces \centerline{% $sM_k(\Gamma) \cap L_k^s(\Gamma \backslash{\cal B})$, } $s \in [1, \infty]$ coincide, where $sM_k(\Gamma)$ denotes the space of super automorphic forms for $\Gamma$ with respect to the weight $k$, and $L_k^s(\Gamma \backslash \cal B)$ denotes the space of $s$-intergrable functions with respect to a certain measure on the quotient $\Gamma\backslash{\cal B}$ depending on $k$. So all these spaces are equal to the space $sS_k(\Gamma) := sM_k(\Gamma)\cap L_k^2(\Gamma\backslash{\cal B}$ of super cusp forms for $\Gamma$ to the weight $k$. \par As it is already well known for automorphic forms on ordinary complex bounded symmetric domains, we will give a proof of this theorem using an unbounded realization $\cal H$ of $\cal B$ and Fourier decomposition at the cusps of the quotient $\Gamma \backslash B$ mapped to $\infty$ via a partial Cayley transformation.

Keywords: Automorphic and cusp forms, complex bounded symmetric domains, super symmetry, semisimple Lie groups, unbounded realization of a complex bounded symmetric domain.

MSC: 11F55; 32C11

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