Automorphic Green Functions Associated with the Secondary Spherical Functions

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

“…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. …”

Green currents for modular cycles in arithmetic quotients of complex hyperballs

“…This can be proved by the same argument as in [Oda and Tsuzuki 2003 (E ; s) be the Fréchet space of all C ∞ -functions f : G → E satisfying the relation f (ma t ng) = e (n+s)t σ (m) f (g) for any m ∈ M 0 , a t ∈ A and n ∈ N . The group G acts by the right translation on the space Ind…”

“…First as in the proof of [Oda and Tsuzuki 2003, Proposition 3.1.1] and then by using the estimations above, we obtain the uniform bound ofP( ϕ s ) for s ∈ U .…”

“…Among other things we show that the value of s at its regular point s = n−2r +2 is square-integrable harmonic form representing the current dd c Ᏻ+δ D up to a constant multiple; see Theorem 7.5. This paper is a continuation of the joint work of Professor Takayuki Oda and the author [Oda and Tsuzuki 2003]. Thanks are due to Professor Takayuki Oda for his interest in this work and fruitful discussions.…”

“…Among many possible choices of the Green currents for a modular cycle Y , this construction may provide a way to fix a natural one. If r = 1, namely Y is a modular divisor, we already obtained a satisfactory result by properly introducing the secondary spherical functions [Oda and Tsuzuki 2003]. So it is quite natural to ask whether the same method works for the higher codimensional case.…”