Rogawski's conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type $A$

The functors constructed by Arakawa and the author relate the representation theory of ${\mathfrak{gl}}_n$ and that of the degenerate affine Hecke algebra $H_{\ell}$ of $\mathrm{GL}_{\ell}$. They transform the Verma modules over ${\mathfrak{gl}}_n$ to the standard modules over $H_{\ell}$. In this paper we prove that they transform the simple modules to the simple modules (in more general situations than in the previous paper). We also prove that they transform the Jantzen filtration on the Verma modules to that on the standard modules. We obtain the following results for the representations of $H_{\ell}$ by translating the corresponding results for ${\mathfrak{gl}}_n$ through the functors: (i) the (generalized) Bernstein-Gelfand-Gelfand resolution for a certain class of simple modules, (ii) the multiplicity formula for the composition series of the standard modules, and (iii) its refinement concerning the Jantzen filtration on the standard modules, which was conjectured by Rogawski.