Abstract We prove that, when n goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product HₓHᵧ H ̲ ^ x H ̲ y of elements of the dual and the usual Kazhdan-Lusztig bases in the Hecke algebra of the symmetric group Sₙ S n stabilizes. As an application, we define the action of projective functors on the principal block of category O O for sl_ sl ∞ and show that the subcategory of finite length objects is stable under this action. As a bonus, we also prove that this latter block is Koszul, answering, for this block, a question from Math. J. 19 (4), 655–693 (2019).
Creedon et al. (Thu,) studied this question.