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We construct an exact functor from the category of Harish-Chandra modules of$\mathrm{GL}_n(\mathbb C)$ to the category of finite-dimensional modules ofgraded Hecke algebras of type A. We show that the functor preservesparabolically induced modules, standard modules, irreducible modules, unitarymodules and Dirac series. We also use the functor to connect aBernstein-Zelevinsky type functor for graded Hecke algebra side to the tensorproduct for $\mathrm{GL}_n(\mathbb C)$ side. Some applications are alsodiscussed.

On the Lefschetz Principle for $\mathrm{GL}(n,\mathbb{C})$ and $\mathrm{GL}(m,\mathbb{Q}_p)$