Koecher-Maa? Series of the Adelic Hermitian Eisenstein Series and the Adelic Hermitian Ikeda Lift for U(m, m)

摘要

Let K = Q(√—D) be an imaginary quadratic field with discriminant —D and with class number h, and x the Dirichlet character corresponding to the extension K/Q. Let m = 2n or 2n + 1 with n a positive integer. Let f be a primitive form of weight 2k + 1 and character x for Γ_0(D), or a primitive form of weight 2k for SL_2(Z) according as m = 2n, or m = 2n + 1. For such an f let Lift(m) (f) be the lift of f to the space of automorphic forms for U(m, m) constructed by Ikeda, and Im(f)_i be its i-th component for 1 < i < h. First we give an explicit formula for the Koecher-Maass series of I_m(f)i. This gives a refined version of our previous result of [Kat14]. As a consequence, we give an explicit formula for the Koecher-Maass series L(s, Lif t~(m)(f)) of Lift~(m)(f). This gives an adelic version of our previous result of [Kat14]. We also give an explicit formula for the Koecher-Maass series of the adelic Hermitian Eisenstein series.