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We prove that the number of indecomposable vector bundles of fixed rank r anddegree d over a smooth projective curve X defined over a finite field is givenby a polynomial (depending only on the pair (r,d) and the genus g of X) in theWeil numbers of X. We provide a closed formula -expressed in terms ofgenerating series- for this polynomial. We also show that the same polynomialcomputes the number of points of the moduli space of stable Higgs bundles ofrank r and degree d over X (at least for large characteristics). This entails aclosed formula for the Poincar\'e polynomial of the moduli spaces of stableHiggs bundles over a compact Riemann surface, and hence also for the Poincar\'epolynomials of the character varieties for the groups GL(r).

Indecomposable vector bundles and stable Higgs bundles over smooth projective curves