Recently, extensive calculations have been made on b-functions of prehomogeneous vector spaces with reducible representations. By examining the results of these calculations, we observe that b-functions of a large number of reducible prehomogeneous vector spaces have decompositions which seem to be correlated to the decomposition of representations. In the present paper, we show that such phenomena can be ascribed to a certain multiplicity one property for group actions on polynomial rings. Furthermore, we give some criteria for the multiplicity one property. Our method can be applied equally to non-regular prehomogeneous vector spaces.

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In the present paper, by using equivariant maps, we construct explicitly relative invariants of the 2-simple prehomogeneous vector spaces of type I, which are classified in [T. Kimura, S. Kasai, M. Inuzuka, O. Yasukura, A classification of 2-simple prehomogeneous vector spaces of type I, J. Algebra 114 (1988) 369-400]. This is a continuation of our previous work [T. Kogiso, G. Miyabe, M. Kobayashi, T. Kimura, Non-regular 2-simple prehomogeneous vector spaces of type I and their relative invariants, J. Algebra 251 (2002) 27-69; T. Kogiso, G. Miyabe, M. Kobayashi, T. Kimura, Relative invariants of some 2-simple prehomogeneous vector spaces, Math. Comp. 72 (2003) 865-889].

We study the b-functions of relative invariants of the prehomogeneous vector spaces associated with quivers of type A. By applying the decomposition formula for b-functions, we determine explicitly the b-functions of one variable for each irreducible relative invariant. Moreover, we give a graphical algorithm to determine the b-functions of several variables.

We investigate the relations for L-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and halfintegral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of SL(2). Our goal is converse theorems for automorphic distributions and Maass forms of level N characterizing them by analytic properties of the associated L-functions. As an application of our converse theorems, we construct Maass forms from the two-variable zeta functions related to quadratic forms defines an automorphic distribution for the group ∆(N ) = ñ(1),ñ(N ) , and hence its Poisson transform, which is given by F α (z) in (III), is a Maass form automorphic for ∆(N ). The transformation formula (0.3) is derived from the comparison of the Poisson transforms of both sides of the summation formula.The group ∆(N ) is a subgroup of the lift Γ 0 (N ) of Γ 0 (N ) to G. In §4 we prove a converse theorem that gives a condition for the automorphic distribution for ∆(N ) associated with the Ferrar-Suzuki summation formula to be automorphic for the larger group Γ 0 (N ) in terms of twists of the corresponding L-functions by Dirichlet characters (Theorem 4.2). By the Poisson transformation, this yields immediately a converse theorem for Maass forms for Γ 0 (N ) (Theorem 4.3), which is an analogue of the converse theorem of Weil [49] for holomorphic modular forms of integral weight (in the case of even ℓ) and its generalization by Shimura [40, Section 5] to holomorphic modular forms of half-integral weight (in the case of odd ℓ). A merit of our approach is to keep calculations involving the Whittaker function to a minimum. The information on the Whittaker function we need is only the fact that the Whittaker function appears as the Fourier transform of the Poisson kernel (see (2.14)). We say now a few words about the assumptions in the converse theorem. In our converse theorem some analytic properties (functional equations, location of poles and so on) are assumed for the L-functions twisted by Dirichlet characters of prime modulus including the principal characters. In the original converse theorem of Weil for holomorphic modular forms ([49]), the assumptions are imposed for the L-functions twisted only by primitive Dirichlet characters of prime modulus. As pointed out in Gelbart-Miller [9, §3.4], it has been considered difficult to transfer the argument of Weil to the Maass form case. In [2] and [3], Diamantis and Goldfeld proved a converse theorem for double Dirichlet series by considering the twists of Dirichlet series by Dirichlet characters including imprimitive characters (the principal characters), namely by the method originally found by Razar [30] for holomorphic modular forms. We follow the approach of Razar and Diamantis-Goldfeld. In a paper [25] that appeared very recently, Nuerurer and Oliver succeeded in modifying the argument of Weil to get a c...