  Suppose that M is a complete hyperkahler manifold with a compact Lie group K
acting via hyperkahler isometries and with hyperkahler moment map
$(\mu_{\mathbb{C}}, \mu_{\mathbb{R}}): M\rightarrow
\mathfrak{k}^*\otimes\operatorname{Im}(\mathbb{H})$. It is a long-standing
problem to determine when the hyperkahler Kirwan map $H^*_K(M,\mathbb{Q})
\longrightarrow H^*(M//K, \mathbb{Q})$ is surjective. We show that for each
$n\geq 2$, the natural $U(n)$-action on $M = T^*(SL_n\times\mathbb{C}^n)$
admits a hyperkahler quotient for which the hyperkahler Kirwan map fails to be
surjective. As a tool, we establish a ``Kahler $=$ GIT quotient'' assertion for
products of cotangent bundles of reductive groups, equipped with the Kronheimer
metric, and representations.

Algebraic Geometry