Math. Ann. 289,663-681 (1991) 9 Springer-Verlag 1991 The Maafl spaces on the Hermitian half-space of degree 2 Aioys Krieg Mathematisches Institut, Westf/ilischeWilhelms-Universit/it, Einsteinstrasse 62, W-4400 Miinster, Federal Republic of Germany Received August 27, 1990 Introduction In 1979 Maal3 [19] introduced the so-called "Spezialschar", which is related with the Saito-Kurokawa conjecture. Later on [20] he investigated the analogue for Siegel modular forms of degree 2 with respect to the non-trivial multiplier system. Similar results for Hermitian modular forms with respect to the Gaussian number field were obtained by Kojima [13], Gritsenko [8] and recently by Raghavan and Sengupta [24]. The general case was dealt with by Sugano [28]. Concerning the half-space of quaternions of degree 2 we refer to [15] and [17]. In this paper we deal with Hermitian modular forms of degree 2 and weight k with respect to the ring of integers o of an imaginary quadratic number field K in the sense of Hel Braun [3]. The attached Maal3 space ~//(k; o) is defined to be the analogue of the Spezialschar in the sense that the Fourier coefficients only depend on the g.c.d, of the entries and the determinant of the attached matrix. It should be mentioned that our definition in general differs from Sugano's [28], who used the isomorphic image of the space of Jacobi forms of index 1 for his definition. If we consider the Fourier Jacobi expansion of a Hermitian modular form in Mr(k; o), we obtain modular forms in Gk- I(DK, ZK), which is the space of elliptic modular forms (of nebentype in Hecke's sense) of weight k- 1 and character X~ on Fo(DK), where - Dr is the discriminant and XK the Kronecker symbol of K. We can avoid the vector valued modular forms in [28] and obtain an isomorphism between the Maal3 space Mr(k; o) and a certain subspace of Gk- I(DK, )~x). This subspace can be viewed as an analogue of Kohnen's "+"-space for modular forms of half-integral weight in [11] and [12] and can be described as the intersection of certain eigenspaces of particular Hecke operators. The isomorphism turns out to be compatible with Hecke operators. As an application we can conclude that the Hermitian Eisenstein series of degree 2 belongs to the Maal3 space. This leads to an explicit formula for the Fourier coefficients of the Hermitian Eisenstein series, which was recently given by Nagaoka [22] in the case that the class number of K is equal to 1. Surprisingly the