Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / A representation of a quiver [Gamma] over a commutative ring R assigns an R-module to each vertex and an R-linear map to each arrow. In this dissertation, we consider R = k[t]/(t[superscript]n) and all R-free representations of [Gamma] which assign a free R-module to each vertex. The category, denoted by Rep[superscript]f[subscript] R([Gamma]), containing all such representations is not an abelian category, but rather an exact category. In this dissertation, we firstly study the Hall algebra of the category Rep[superscript]f[subscript] R([Gamma]), denote by [Eta](R[Gamma]), for a loop-free quiver [Gamma]. A geometric realization of the composition subalgebra of [Eta](R[Gamma]) is given under the framework of Lusztig's geometric setting. Moreover, the canonical basis and a monomial basis of this subalgebra are constructed by using perverse sheaves. This generalizes Lusztig's result about the geometric realization of quantum enveloping algebra. As a byproduct, the relation between this subalgebra and quantum generalized Kac-Moody algebras is obtained. If [Gamma] is a Jordan quiver, which is a quiver with one vertex and one loop, each representation in Rep[superscript]f[subscript] R([Gamma]), gives a matrix over R when we fix a basis of the free R-module. An interesting case arises when considering invertible matrices. It then turns out that one is dealing with representations of the group GL[subscript]m(k[t]/(t[superscript]n)). Character sheaf theory is a geometric character theory of algebraic groups. In this dissertation, we secondly construct character sheaves on GL[subscript]m(k[t]/(t[superscript]2)). Then we define an induction functor and restriction functor on these perverse sheaves.

Pour toute courbe projective lisse C et théorie homologique orientée de Borel-Moore libre A, on construit un produit associatif de type Hall sur les A-groupes du champ de modules des faisceaux de Higgs de torsion sur C.On montre que l'algèbre AHa0C qu'on obtient admet une présentation de battage naturelle, qui est fidèle dans le cas où A est l'homologie de Borel-Moore usuelle.On introduit de plus les espaces de modules des triplets stables M(d,n), fortement inspirés par les variétés de carquois de Nakajima.Ces espaces de modules sont des variétés lisses symplectiques, et admettent une autre caractérisation comme les espaces de modules de faisceaux sans torsion stables encadrés sur P(T*C)$.De plus, on munit leurs A-groupes avec une action de AHa0C, qui généralise les opérateurs de modification ponctuelle de Nakajima sur l'homologie des schémas de Hilbert de T*C. / For a smooth projective curve C and a free oriented Borel-Moore homology theory A, we construct a Hall-like associative product on the A-theory of the moduli stack of Higgs torsion sheaves on C.We show that the resulting algebra AHa0C admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel-Moore homology groups.We also introduce moduli spaces of stable triples M(d,n), heavily inspired by Nakajima quiver varieties.These moduli spaces are shown to be smooth symplectic varieties, which admit another characterization as moduli of framed stable torsion-free sheaves on P(T*C).Moreover, we equip their A-theory with an AHa0C-action, which generalizes Nakajima's raising operators on the homology of Hilbert schemes of points on T*C.