Parahoric restriction and the relationship between local theorems

Determine whether parahoric restriction relates Theorem 1 (the non-archimedean local identity for twisted Jacquet modules) and Theorem 2 (the finite-field decomposition of the corresponding twisted Jacquet module) for depth-zero inducing data.

Background

The paper proves two parallel results for twisted Jacquet modules associated with representations induced from Speh representations. Over a non-archimedean local field of characteristic zero, Theorem 1 identifies the maximal G-coinvariant quotient, outside a finite exceptional parameter set, with an induced representation involving a shorter Speh representation and the conjugate of an admissible representation of G. Over finite fields of sufficiently large characteristic, Theorem 2 gives a complete multiplicity-free decomposition under additional hypotheses, including cuspidality of the inducing representation and a rank condition.

The authors note that depth-zero inducing data provide a setting in which these two results might be compared through parahoric restriction, a process relating representations over p-adic groups to representations of finite reductive quotients. The unresolved issue is whether parahoric restriction actually connects the characteristic-zero local identity with the finite-field decomposition established in the paper.