Twisted Aubert duality: involution and functorial compatibilities

Determine whether the twisted Aubert duality functor π ↦ π on the category Rep(G) of smooth finite-length representations of the disconnected group G = G ⋊ θ, defined in Appendix B.4 via the complex X·(π) and the maps Xt(θ), is an involution, and establish whether this twisted Aubert duality functor commutes with the contragredient functor, parabolic induction functors, and Jacquet functors.

Background

Appendix B develops a twisted version of Aubert duality for disconnected reductive groups G = G ⋊ θ, extending the standard Aubert involution from connected groups. The construction uses complexes built from parabolic induction and Jacquet functors and an equivariant action of the twisting automorphism θ.

While the standard Aubert duality for connected groups is known to be an involution and to satisfy expected compatibilities, the authors note that analogous properties for the twisted setting are not established in this work. They anticipate resolving these properties in a forthcoming paper, but do not rely on them here.