Math. Ann. 276, 675-686 (1987) Am 9 Spdnger-Verlag 1987 The Maafl-Spaee on the Half-Space of Quaternions of Degree 2 Aloys Krieg MathematischesInstitut, Einsteinstrasse62, D-4400 Miinster, Federal Republic of Germany Introduction In 1979 MaaB [9-11] investigated the so-called "Spezialschar", which is related with the Saito-Kurokawa-conjecture (cf. [8] or [13]). Analogous results in the space of Hermitian modular forms with respect to the Gaussian integers were obtained by Kojima [5] and recently by Gricenko [3]. In the present paper, the MaaB-space ,//(k; R-I) is introduced as the analogue of the "Spezialschar" on the half-space of quaternions of degree 2. The theory of modular forms of quaternions was pointed out in [61 from which the notations are adopted. In order to determine JC(k;I-1), the same methods as in [9-11] are applied. Surprisingly the results become simpler than in the description of the "Spezialschar". Considering the Fourier-Jacobi-expansion of modular forms in ~r ~,I), elliptic modular forms with respect to a congruence subgroup of level 2 instead of 4 and to the trivial multiplier system arise, their weight turns out to be even again instead of half-integral. The dimension of the arising space of elliptic modular forms can be calculated explicitly. This leads to the astonishing result that the MaaB-space v/t'(k;R-l) consists exactly of the liftings of the "Spezialschar". In contrast to this the MaaB- space on the Hermitian half-space, which was investigated in [5], has a greater dimension than the "Spezialschar", whenever the weight k satisfies k> 8 and k=0 mod4. 1. The Definition of the Maafl-Space .4t'(k; IF) Let F stand for •, C or H as is the case in [6]. The attached order is denoted by 0 = d~(F), i.e. d~fR)= Z , d~(IE) =Zel +Ze2 (Gaussian integers), 9 (R-I)=Ze0 +~-el +7.e2 +Ze3 (quaternions of Hurwitz),