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We extend the methods of geometric invariant theory to actions of non–reductive groups
in the case of homomorphisms between decomposable sheaves whose automorphism groups are
non–reductive. Given a linearization of the natural action of the group on Hom(E,F), a homomorphism is called stable if its
orbit with respect to the unipotent radical is contained in the stable locus with respect
to the natural reductive subgroup of the automorphism group. We encounter effective
numerical conditions for...
Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely...
In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Muñoz. Using these results we obtain an explicit formula for the non-abelian zeta functions,...
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