A Satake type theorem for Super Automorphic forms

TL;DR

This paper establishes a Satake type theorem for super automorphic forms on a rank 1 complex bounded symmetric super domain, linking super cusp forms with finiteness of p-norms across all p, and showing the equivalence of these Lp-spaces.

Contribution

It proves a Satake type theorem for super automorphic forms, connecting cusp forms with p-norm finiteness and demonstrating the coincidence of all Lp-spaces in this context.

Findings

Super automorphic forms are super cusp forms if and only if their p-norms are finite for all p.

All Lp-spaces of super automorphic forms on the domain coincide.

The proof uses an unbounded realization and Fourier decomposition at cusps.

Abstract

Aim of this article is a Satake type theorem for super automorphic forms on a complex bounded symmetric super domain B of rank 1 with respect to a lattice. This theorem - roughly speaking - says that for large weight k and all p from 1 to infinity (both including) a super automorphic form on B is a super cusp form if and only if its p-norm with respect to a certain measure on the quotient of B is finite. And so in particular all these Lp-spaces coincide! We will give a proof of this theorem using an unbounded realization of B and Fourier decomposition at the cusps of the quotient mapped to infinity via a partial Cayley transform.