Representation spaces of quivers play a preeminent role in geometric representation theory. For example, starting from them one can construct algebraic stacks whose K-theory has an associative algebra structure (K-theoretic Hall algebra), and smooth quasi-projective varieties (called Nakajima quiver varieties) whose K-theory provides a representation of these algebras.
In the first part of this talk, I will review the theory of Nakajima quiver varieties and K-theoretic Hall algebras associated with a quiver. Then I will explain what kind of new results one can obtain by replacing the representations spaces by the category of coherent sheaves over the complex projective line. (Joint work with O. Schiffmann.)