An explicit formula of the Koecher-Maass series associated with Hermitian modular forms belonging to the Maass space
Let \(H\) denote the space of \(2\times 2\) complex matrices \(Z\) such that \(i({}^t \bar{Z}-Z)\) is positive definite. \(H\) is equivalent to a bounded symmetric domain belonging to E. Cartan's first type. The group is \(\text{U}(2,2)\). Arithmetic subgroups can be defined using the integers of an imaginary quadratic field. We then have the notion of modular form on \(H\) of weight \(k\) (these functions are called hermitian modular forms of degree 2 and weight \(k\); they are holomorphic). In [\textit{A. Krieg}, Math. Ann. 289, 663--681 (1991; Zbl 0713.11033)] a lift is defined from a certain space of modular forms in one variable of weight \(k-1\) (with a multiplier) to hermitian modular forms of degree 2 and weight \(k\). This lift is injective and its image is called Maass space, because it is an analogue of a space of Siegel modular forms of degree 2 introduced by \textit{H. Maaß} [Invent. Math. 52, 95--104 (1979; Zbl 0386.10013); 53, 249--265 (1979; Zbl 0413.10021)]. To any hermitian modular form one can attach a Dirichlet series using the Fourier coefficients of the form (it is called Koecher-Maass series, because Maass and Koecher studied the same for Siegel modular forms). Now let \(F\) be a function in the Maass space and let \(f\) be the modular form of one variable corresponding to \(F\). It is natural to ask if there is an expression of the Koecher-Maass series of \(F\) in terms of \(L\)-functions related to \(f\). This question is answered in the present paper. The result is an explicit expression of the Koecher-Maass series as a finite sum of products of two \(L\)-functions attached to twists of \(f\) by a character, when \(f\) is an eigenfunction of the Hecke operators. It would be interesting to study the lift considered here from the representation-theoretic point of view.
scientific article; zbMATH DE number 5013739