The pure cohomology of multiplicative quiver varieties

To a quiver Q and choices of nonzero scalars q_i, non-negative integers \alpha _i, and integers \theta _i labeling each vertex i, Crawley-Boevey–Shaw associate a multiplicative quiver variety {\mathcal {M}}_\theta ^q(\alpha ), a trigonometric analogue of the Nakajima quiver variety associated to Q, \alpha , and \theta . We prove that the pure cohomology, in the Hodge-theoretic sense, of the stable locus {\mathcal {M}}_\theta ^q(\alpha )^{{\text {s}}} is generated as a {\mathbb {Q}}-algebra by the tautological characteristic classes. In particular, the pure cohomology of genus g twisted character varieties of GL_n is generated by tautological classes.