Automorphic Green functions associated with the secondary spherical functions (Q1423885)

Let \(G\) be a connected semisimple algebraic group of Hermitian type defined over \(\mathbb Q\), and let \(K\) be a maximal compact subgroup of \(G_\mathbb R\). Then \(\mathcal D = G_\mathbb R /K\) is a bounded symmetric domain, and an arithmetic subgroup \(\Gamma\) of \(G\) determines the corresponding arithmetic variety \(\Gamma \backslash \mathcal D\). Let \(H\) be a reductive subgroup of \(G\) such that \(H_\mathbb R \cap K\) is a maximal compact subgroup of \(H_\mathbb R\) and \(\mathcal D_0 = H_\mathbb R \cap K \backslash H_\mathbb R\) is a subdomain of \(\mathcal D\) of complex codimension one. In this paper the authors introduce a spherical function \(\phi_s\) on \(G\) for \(s \in \mathbb C\) and define a Poincaré series \(\mathcal P_s\) on \(\Gamma \backslash \mathcal D\) as an infinite series involving \(\phi_s\). They prove that \(\mathcal P_s\) is meromorphic and is a fundamental solution of a Poisson equation. They also give a proof of the meromorphic continuation of \(\mathcal P_s\) with respect to \(s\) and show that \(\mathcal P_s\) satisfies an expected property of a Green current of logarithmic type.