  We introduce the notion of filtered representations of quivers, which is
related to usual quiver representations, but is a systematic generalization of
conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices
up to conjugation by invertible (block) upper triangular matrices. With this
notion in mind, we describe the ring of invariant polynomials for interesting
families of quivers, namely, finite $ADE$-Dynkin quivers and affine type
$\widetilde{A}$-Dynkin quivers. We then study their relation to an important
and fundamental object in representation theory called the
Grothendieck-Springer resolution, and we conclude by stating several
conjectures, suggesting further research.
