Taku Ishii

<p>Let and be positive integers such that . Let be either or . Let and be maximal compact subgroups of and , respectively. We give the explicit descriptions of archimedean Rankin–Selberg integrals at the minimal - and -types for pairs of principal series representations of and , using their recurrence relations. Our results for can be applied to the arithmetic study of critical values of automorphic -functions.</p>

We consider here the archimedean zeta integrals for GL(3) x GL(2) and show that the zeta integral for appropriate Whittaker functions is equal to the associated L-factor.

We study Whittaker functions for the principal series representations of SL(n,R). We derive a system of partial differential equations characterizing our Whittaker functions. We give explicitly power series solutions at the regular singularity of the system, and integral representations of the unique moderate growth Whittaker functions. (C) 2013 Elsevier Inc. All rights reserved.

In this paper, we evaluate archimedean zeta integrals for automorphic L-functions on GL(n) x GL(n-1+l) and on SO2n+1 x GL(n+l), for l = -1, 0, and 1. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals. When l = 0 or l = 1, the archimedean zeta integrals amount to integrals over the full torus. We show that, as has been predicted by Bump for such domains of integration, these zeta integrals are equal to the corresponding local L-factors-which are simple rational combinations of Gamma functions. (In the cases of GL(n) x GL(n-1) and GL(n) x GL(n) this has, in large part, been shown previously by the second author of the present work, though the results here are more general in that they do not require the assumption of trivial central characters. Our techniques here are also quite different. New formulas for GL(n, R) Whittaker functions, obtained recently by the authors of this work, allow for substantially simplified computations). In the case l = -1, we express our archimedean zeta integrals explicitly in terms of Gamma functions and certain Barnes-type integrals. These evaluations rely on new recursive formulas, derived herein, for GL(n, R) Whittaker functions. Finally, we indicate an approach to certain unramified calculations, on SO2n+1 x GL(n) and SO2n+1 x GL(n+1), that parallels our method herein for the corresponding archimedean situation. While the unramified theory has already been treated using more direct methods, we hope that the connections evoked herein might facilitate future archimedean computations.

We give explicit formulas for Whittaker functions for the class one principal series representations of the orthogonal groups SO2n+1(R) of odd degree. Our formulas are similar to the recursive formulas for Whittaker functions on SLn(R) given by Stade and the author.

We introduce the explicit formulas of archimedean Whittaker functions on GL(3) and their application to archimedean zeta integrals.

We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.

Let G = GSp( 2) be the symplectic group with similitude of degree two, which is defined over Q. For a generic cusp form F on the adelized group G(A) whose archimedean type is a principal series representation, we show that its spinor L-function is continued to an entire function and satisfies the functional equation.

In this paper, we give explicit formulas for the secondary and the primary Whittaker functions for P-J-principal series representations of Sp(3, R). (c) 2007 Elsevier Inc. All fights reserved.

We derive new recursive formulas for principal series Whittaker functions, of both fundamental and class one type, on GL(n, R). These formulas relate such Whittaker functions to their counterparts on GL(n - 1, R), instead of on GL(n - 2, R) as has been the case in numerous earlier studies. In the particular case n = 3, and for certain special values of the eigenvalues, our formulas are seen to resemble classical summation formulas, due to Gegenbauer, for Bessel functions. To illuminate this resemblance we also derive, in this work, formulas for certain special values of GL(3, R) fundamental Whittaker functions. These formulas may be understood as analogs of expressions derived by Bump and Friedberg for class one Whittaker functions on GL(3, R). (c) 2007 Elsevier Inc. All rights reserved.

We discuss a confluence from Siegel-Whittaker functions to Whittaker functions on Sp(2, R) by using their explicit formulae. In our proof, we use expansion theorems of the good Whittaker functions by the secondary Whittaker functions.

We give explicit formulas for principal series Whittaker functions on Sp(2,R). (C) 2005 Elsevier Inc. All rights reserved.

We prove the recursive integral formula of class one M-Whittaker functions on SL(n, R) conjectured and verified in case of n = 3, 4 by Stade.

In this paper we study a kind of spherical function, which we call a Siegel-Whittaker function, related to Fourier expansions of automorphic forms on Hermitian symmetric domains of type IV. We obtain a multiplicity-one theorem and an integral representation of this spherical function.