Counterexamples to hyperkahler Kirwan surjectivity

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, $\mathbb{Q}$)$issurjective\mathrm{.}Weshowthatforeach$n\geq 2$,thenatural$U(n)$-actionon$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.