We identify the double dual EPW sextic Y A ⊥ and the double EPW sextic Y A , associated with a very general Gushel-Mukai fourfold X, with the Bridgeland moduli spaces of stable objects of character Λ 1 and Λ 2 in the Kuznetsov component Ku(X). This provides an affirmative answer to a question of Perry-Pertusi-Zhao. As an application, we prove a conjecture of Kuznetsov-Perry for very general Gushel-Mukai fourfolds.

We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperkähler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel-Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperkähler varieties.