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The notion of derivatives for smooth representations of $GL(n,\mathbb{Q}_p)$was defined in [BZ77]. In the archimedean case, an analog of the highestderivative was defined for irreducible unitary representations in [Sah89] andcalled the "adduced" representation. In [AGS] derivatives of all orders weredefined for smooth admissible Frechet representations (of moderate growth).  A key ingredient of this definition is the functor of twisted coinvariantswith respect to the nilradical of the mirabolic subgroup. In this paper weprove exactness of this functor and compute it on a certain class ofrepresentations. This implies exactness of the highest derivative functor, andallows to compute highest derivatives of all monomial representations.  In [AGS] these results are applied to finish the computation of adducedrepresentations for all irreducible unitary representations and to proveuniqueness of degenerate Whittaker models for unitary representations.

Twisted homology for the mirabolic nilradical