A Fourier–Jacobi Dirichlet series for cusp forms on orthogonal groups

We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms <i>F</i>, <i>G</i> for orthogonal groups of signature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo></mml:mrow> </mml:math> . In the case when <i>F</i> is a Hecke eigenform and <i>G</i> is a Maass lift of a Poincaré series, we establish a connection with the standard <i>L</i>-function attached to <i>F</i>. What is more, we find explicit choices of orthogonal groups, for which we obtain a clear-cut Euler product expression for this Dirichlet series. Through our considerations, we recover a classical result for Siegel modular forms, first introduced by Kohnen and Skoruppa, but also provide a range of new examples, which can be related to other kinds of modular forms, such as paramodular, Hermitian, and quaternionic.