The Secondary Spherical Functions and Automorphic Green Currents for Certain Symmetric Pairs

Abstract

Given a modular embedding j : ∆\H/H ∩ K → Γ\G/K associated with an equivariant embedding (H, H/H ∩K) → (G,G/K) of symmetric domains with actions of a semisimple Lie group G and a reductive subgroup H, both defined over Q compatibly, together with some other conditions. Starting from a certain harmonic left H-invariant “spherical” current on G/K with sigularity along HK/K, we can define a Poincare series. Apply ∂∂ operator to the analytic continutation of this with respect to the parameter of eigenvalues of the “Laplacian”. Then as an analogue of the Kronecker limit formula, we can construct a Green current for the cycle defined by j. This is a continuation of the previous paper [26], and here we treat the case of higher-codimensional cycles with compact Γ\G/K.