Generalizing the Distinguished Lift Property to other involutions and inner forms (l ≠ 2)

Establish that for any involution θ of GL_n(F) and, more generally, for any involution of any inner form G of GL_n(F), and for any prime l ≠ 2, a cuspidal irreducible F̄_l-representation π of G is H-distinguished (where H = G^θ) if and only if π admits an integral Q_l-lift \tilde{π} of G such that \tilde{π} is H-distinguished. Here, H-distinguished means Hom_H(π, 1) ≠ 0, and a Q_l-lift is an integral Q_l-representation whose reduction mod l is isomorphic to π.

Background

Theorem 1.2 of the paper proves a Distinguished Lift Property for GL_n(F) with H = GL_n(F_0) arising from the Galois involution associated to a quadratic extension F/F_0, in the case l ≠ 2: a cuspidal F̄_l-representation is H-distinguished if and only if it has an H-distinguished cuspidal Q_l-lift.

The authors suggest extending this phenomenon beyond the specific Galois involution to arbitrary involutions of GL_n(F) and, more broadly, to involutions of inner forms of GL_n(F). Such a generalization would substantially broaden the scope of the classification of distinguished l-modular cuspidal representations via lifting.

They note potential obstructions for reductive groups beyond general linear groups, but indicate optimism for other involutions within GL_n(F) and its inner forms.