Dirichlet series and automorphic forms on unitary groups

Abstract:

In a special case our unitary group takes the form \[ G = \{ g \in {\text {GL}}(p + 2,{\mathbf {C}}){|^t}\bar gRg = R\} .\] Here \[ R = \left ( {\begin {array}{*{20}{c}} S & 0 & 0 \\ 0 & 0 & 1 \\ 0 & { - 1} & 0 \\ \end {array} } \right )\] is a skew-Hermitian matrix with entries in an imaginary quadratic number field $K$. We suppose that $- iR$ has signature $(p + 1,1)$. This group acts naturally on the symmetric domain \[ D = \left \{ {w \in {{\mathbf {C}}^p},z \in {\mathbf {C}}|\operatorname {Im} (z) > - {{\frac {1}{2}}^t}\bar wSw} \right \}.\] If $\Gamma = G \cap {\text {SL}}(p + 2,{\mathcal {O}_K})$ with ${\mathcal {O}_K}$ the ring of integers in $K$, then an automorphic form $f(w,z)$ with respect to $\Gamma$ has an expansion ${\Sigma _r}{g_r}(w) \cdot {e^{2\pi irz}}$. The functions ${g_r}(w)$ are theta functions. Given another automorphic form $g(w,z)$ with an expansion ${\Sigma _s}{h_s}(w) \cdot {e^{2\pi isz}}$ we define a Dirichlet series ${\Sigma _r}\langle {g_r},{h_r}\rangle {r^{ - s}}$ . Here $\langle {g_r},{h_r}\rangle$ is a certain positive definite inner product on the space of theta functions. The series is obtained as an integral of Rankin type: \[ {\int _{{P_\Gamma }\backslash D}}f\bar g\cdot {\left ( {\operatorname {Im} (z) + \frac {1} {2}{i^t}\bar wSw} \right )^s}dw\;d\bar w\;dz\;d\bar z\] with ${P_\Gamma } \subseteq \Gamma$ a subgroup of "translations". The series is analytically continued by studying the Eisenstein series arising when the above integral is transformed into an integral over $\Gamma \backslash D$. In the case $p = 1$ our results have an application to some recent work of Shintani, where the Euler product attached to an eigenfunction of the Hecke operators is obtained, up to some simple factors, as a series of the above type.