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.