On the double EPW sextic associated to a Gushel-Mukai fourfold

In analogy to the case of cubic fourfolds, we discuss the conditions under which the double cover ilde{Y}_A of the EPW sextic hypersurface associated to a Gushel-Mukai fourfold is birationally equivalent to a moduli space of (twisted) stable sheaves on a K3 surface. In particular, we prove that ilde{Y}_A is birational to the Hilbert scheme of two points on a K3 surface if and only if the Gushel-Mukai fourfold is Hodge-special with discriminant d such that the negative Pell equation P_{d/2}(-1) is solvable in mathbb{Z}.