On modular Harish-Chandra series of finite unitary groups

In the modular representation theory of finite unitary groups when the characteristic ℓ \ell of the ground field is a unitary prime, the s l ^ e \widehat {\mathfrak {sl}}_e -crystal on level 2 2 Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for ℓ \ell sufficiently large, the s l ∞ \mathfrak {sl}_\infty -crystal on the same level 2 2 Fock spaces provides the remaining piece of the puzzle for the full Harish-Chandra branching rule.