A study on relations between the theory of prehomogeneous vector spaces, the theory of group representations and the theory of automorphic forms

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In this research project, we investigated zeta functions of prehomogeneous vector spaces from the view points(1)relations between functional equations and representations of general linear groups,(2)relations to automorphic L-functions,(3)generalization of the theory to non-regular prehomogeneous vector spaces.(1)We showed that the functional equations of zeta functions are closely related to intertwining operators between degenerate principal series representations of general linear groups, and, using the relation, we obtained an integral expression of Eulerian type of the gamma matrices of functional equations. This enables us to identify the variable change in functional equations as an action of an element in the Weyl group of a general linear group, and to decompose functional equations into a product of more elementary functional equations. There exists a similar results for p-adic local zeta functions. As an application of p-adic theory, we investigated the Fourier coefficients … More of Elsenstein series of Sp(n) and GL(n) and the theory of spherical transforms on certain spherical homogeneous spaces.(2)We identified the Koecher-Maass series of real analytic Siegel Eisenstein series with a zeta function associated with a certain prehomogeneous vector space on which the Siegel parabolic subgroup of SO(n, n) acts. It is quite probable that this result can be extended to other classical groups. A considerable progress has been made in explicit calculation of zeta functions. We obtained an explicit expression of zeta functions in terms of the Riemann zeta function and the Mellin transforms of the Cohen Eisenstein series for more than 70 percent of irreducible regular reduced prehomogeneous vector spaces.(3)For non-regular prehomogeneous vector spaces, we developed a general theory of integral representations and the functional equation of the zeta integrals, which is a formal generalization of the theory for regular prehomogeneous vector spaces. We also gave the first example of explicit functional equations for non-regular spaces. Less