Graham type theorem on classical bounded symmetric domains

Graham Theorem on the unit ball B n in C n states that every invariant harmonic function u ∈ C n ( B ¯ n ) must be pluriharmonic in B n (Graham in Commun Partial Differ Equ 8(5):433–476, 1983 ). This rigidity phenomenon of Graham has been studied by many authors [see, for examples, Graham and Lee (Duke Math J 57:697–720, 1988 ), Li and Simon (Am J Math 124:1045–1057, 2002 ), Li and Wei (Sci China Math 53:779–790, 2010 ), etc]. In this paper, we prove that Graham theorem holds on classical bounded symmetric domains, which include Type I and Type II domains, Type III domain III(n) and Type IV domain IV( n ) with even n ≥ 4 .