Generalised Prym Varieties as Fixed Points

Let X be a nonsingular, projective curve of genus g ≥ 2. Then the elements of order r of the Jacobian of X act in a natural way on the moduli space M(r, ξ) of stable vector bundles on X of rank r( ≥ 2) whose determinants are isomorphic to a given line bundle ξ of degree d. We shall assume that r and d are coprime and show that the fixed point variety corresponding to any element μ (strictly) of order r is an abelian variety. In fact, this fixed point variety is isomorphic to the generalised Prym variety (See Remark 3.7) associated to μ.