Relations between prehomogeneous vector spaces of parabolic type and representation theory

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We have completed the work on b-functions associated quivers of type A, and published a paper in a journal. Applying a result in the paper, we have obtained some results on functional equations of prehomogeneous vector spaces of parabolic type arising from special linear Lie algebras. Moreover, we have studied the L-functions associated with automorphic distributions defined by Toshiaki Suzuki. As a result, we have observed that the functional equation satisfied by the L-functions associated with automorphic distribution coincides with that satisfied by the L-functions associated with Maass wave forms. Furthemore, we have formulated a converse theorem for automorphic distributions, and combining this with Poisson transforms, we have obtained another proof for the converse theorem for Maass wave forms.