都築 正男

Abstract In this paper, we give an explicit formula for the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we obtain a functional equation satisfied by the Shintani double zeta functionsin addition to Shintani’s functional equations.Second, we establish the holomorphicity of a certain Dirichlet series generalizing a result by Ibukiyama and Saito. This Dirichlet series occurs in the study of unipotent contributions of the geometric side of the Arthur–Selberg trace formula of the symplectic group.Third, we prove an asymptotic formula of the weighted average of the central values of quadratic Dirichlet L-functions.

We consider a BSDE (backward stochastic differential equation) % $$\left\{\begin{array}{l} -dY(t)=f(B(\cdot),t,Y(t),Z(t))dt-Z(t)^*dB(t), \\ Y(1)=ξ. \end{array}\right.$$ % We construct backward stochastic difference equations approximating the BSDE, where time and space are discrete. We show the existence and uniqueness of the solutions of the backward stochastic difference equations. Also we show a convergence result of the solutions of the backward stochastic difference equations towards that of the BSDE.

Let $G=\sU(n,1)$ and $H=\sU(n-1,1) × \sU(1)$ with $n\geqslant 2$. We realize $H$ as a closed subgroup of $G$, so that $(G,H)$ forms a semisimple symmetric pair of rank one. For irreducible representations $π$ and $η$ of $G$ and $H$ respectively, we consider the space ${\cal I}_{η,π}={\rm Hom}_{\g_\C,K} (π,{\rm Ind}_H^G(η))$ with $K$ a maximal compact subgroup in $G$ and $\g_\C$ the complexified Lie algebra of $G$. The functions that belong to ${\rm Im}(Φ)$ for some $Φ\in {\cal I}_{η,π}$ will be called the {\it Shintani functions}. We prove that ${\rm dim}_\C{\cal I}_{η,π}\leqslant 1$ for any $π $ and any $η$, giving an explicit formula of the Shintani functions that generate a \lq corner\rq\ $K$-type of $π$ in terms of Gaussian hypergeometric series. We also give an explicit formula of corner $K$-type matrix coefficients of $π$ in the usual sense.

We shall present an explicit formula of ""generalized"" spherical functions on $SU(2,1)$ with respect to its reductive spherical subgroup $S\bigl(U(1,1) × U(1)\bigr)$, which can be considered to be a real analogue of the Whittaker-Shintani functions introduced by Shintani and investigated by Murase and Sugano. At the same time, we shall prove a multiplicity one theorem for the corresponding space of intertwining operators.