Dirichlet series and automorphic forms on unitary groups

Abstract

In a special case our unitary group takes the form \[ G = { g ∈ GL ( p + 2 , C ) | t g ¯ R g = R } . G = \{ g \in {\text {GL}}(p + 2,{\mathbf {C}}){|^t}\bar gRg = R\} . \] Here \[ R = ( S a m p ; 0 a m p ; 0 0 a m p ; 0 a m p ; 1 0 a m p ; − 1 a m p ; 0 ) R = \left ( {\begin {array}{*{20}{c}} S & 0 & 0 \\ 0 & 0 & 1 \\ 0 & { - 1} & 0 \\ \end {array} } \right ) \] is a skew-Hermitian matrix with entries in an imaginary quadratic number field K K . We suppose that − i R - iR has signature ( p + 1 , 1 ) (p + 1,1) . This group acts naturally on the symmetric domain \[ D = { w ∈ C p , z ∈ C | Im ⁡ ( z ) > − 1 2 t w ¯ S w } . D = \left \{ {w \in {{\mathbf {C}}^p},z \in {\mathbf {C}}|\operatorname {Im} (z) > - {{\frac {1}{2}}^t}\bar wSw} \right \}. \] If Γ = G ∩ SL ( p + 2 , O K ) \Gamma = G \cap {\text {SL}}(p + 2,{\mathcal {O}_K}) with