REPRESENTATION VARIETIES OF THE FUNDAMENTAL-GROUPS OF COMPACT NONORIENTABLE SURFACES

Let Gamma(g) be the fundamental group of a compact non-orientable surf ace of genus g and let K be an algebraically closed field of character istic O. The structure of the representation varieties R(Gamma(g), GL( n)(K)), R(Gamma(g),SLn(K)) of Gamma(g) into GL,(K) and SLn(K) and of t he character varieties X(Gamma(g),GL(n)(K)) is described; namely, the number of their irreducible components and their dimensions are determ ined and their birational properties are investigated. It is proved, i n particular, that all the irreducible components of R(Gamma(g),GL(n)( K)) are Q-rational varieties.