Abstract

Using a known recursive formula for the class one principal series GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]) Whittaker function, we deduce a recursive formula for the multiple Mellin transform of this function. From the latter formula, we verify a conjecture of Goldfeld regarding the location of poles of our Mellin transform. We further express the residues at these poles in terms of Mellin transforms of lower-rank Whittaker functions. Our next result concerns the simplification of our Mellin transform under a certain restriction on the transform parameter. We show, by applying a change of variable to our above result on poles of the Mellin transform, that the GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]) transform reduces essentially to a GL(n - 1, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]) transform under this restriction. We then demonstrate that, under further restriction of the Mellin transform parameter, this Mellin transform in fact reduces to a ratio of products of gamma functions. Our result proves a conjecture of Bump and Friedberg that is motivated by the theory of exterior square automorphic L-functions. Finally, we show that a certain Mellin transform of a product of two Whittaker functions (one on GL(n - 1, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]), and the other on GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /])) reduces to a product of gamma functions. This last result verifies a conjecture of Bump regarding archimedean Euler factors of automorphic L-functions on GL(n - 1, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /]) x GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /]).

pdf

Share