Analysis on arithmetic quotients

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One of the major developments in number theory during the 20th century was Robert Langlands' work on automorphic forms, with its ultimate goal of understanding L-functions, a primary tool in analytic number theory ever since they were first invented by Lejeune Dircihlet in the early 19th century. Langlands' first important results in this field was his construction of Eisenstein series, which implied the meromorphic continuation of a new family of L-functions. His work on Eisenstein series has not been subtantially simplified in the nearly 45 years since he first did it, and I hope to both simplify and extend it, looking at it from a new perspective. Langlands' technique gave only meromorphic continuation, but several others, primarily Freydoon Shahidi of Purdue University, have used Eisenstein series to prove stronger properties, including a functional equation, and applied these L-functions to obtain new estimates of certain numerical constants leading to an analogue of Ramanujan's conjecture. In one sense these techniques have come to a dead end, since the list of all relevant finite-dimensional reductive groups has been exhausted. In spite of negative evidence published by Shahidi, I believe the infinite-dimensional Kac-Moody groups are worth exploring.