On two geometric constructions of and its representations

Ginzburg and Nakajima have given two different geometric constructions of quotients of the universal enveloping algebra of and its irreducible finite-dimensional highest weight representations using the convolution product in the Borel–Moore homology of flag varieties and quiver varieties, respectively. The purpose of this paper is to explain the precise relationship between the two constructions. In particular, we show that while the two yield different quotients of the universal enveloping algebra, they produce the same representations and the natural bases which arise in both constructions are the same. We also examine how this relationship can be used to translate the crystal structure on irreducible components of quiver varieties, defined by Kashiwara and Saito, to a crystal structure on the varieties appearing in Ginzburgʹs construction, thus recovering results of Malkin.