LE MATEMATICHE Vol. LV (2000)  Fasc. II, pp. 437446 PROJECTIVE MODULI SPACE OF SEMISTABLE PRINCIPAL SHEAVES FOR A REDUCTIVE GROUP TOM ´ AS L. G ´ OMEZ - IGNACIO SOLS Dedicated to Silvio Greco in occasion of his 60-th birthday. 1. Introduction. This contribution to the homage to Silvio Greco is mainly an announce- ment of results to appear somewhere in full extent, explaining their development from our previous article [5] on conic bundles. In [11] and [15] Narasimhan and Seshadri de�ned stable bundles on a curve and provided by the techniques of Geometric Invariant Theory (GIT) de- veloped by Mumford [10] a projective moduli space of the stable equivalence classes of semistable bundles. Then Gieseker [4] and Maruyama [8] [9] gener- alized this construction to the case of a higher-dimensional projective variety, obtaining again a projective moduli space by also allowing torsion-free sheaves. Ramanathan [12] [13] has provided the moduli space of semistable principal bundles on a connected reductive group G , thus generalizing the Narasimhan and Seshadri notion and construction, which then becomes the particular case G = Gl (n , C). Faltings [3] has considered the moduli stack of principal bundles on semistable curves. For G orthogonal or symplectic he considers a torsion-free Both authors are members of EAGER (EC FP5 Contract no. HPRN-CT-2000- 00099). T. G. was supported by a postdoctoral fellowship of Ministerio de Educaci´ on y Cultura (Spain)