ICERM Semester Program: Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series
L-functions—vast generalizations of the Riemann zeta function— are fundamental objects of study in number theory. In the 1980's the idea emerged that it could be useful to tie together a family of related L-functions in one variable to create a "double Dirichlet series," which could be used to study the average behavior of the original family of L-functions. Double Dirichlet series soon became multiple Dirichlet series. It has gradually emerged that the local structure of these multiple Dirichlet series shows a rich connection to combinatorial representation theory.
This program will explore this interface between automorphic forms and combinatorial representation theory, and will develop computational tools for facilitating investigations. On the automorphic side, Whittaker functions on p-adic groups and their covers are the fundamental objects. Whittaker functions and their relatives are expressible in terms of combinatorial structures on the associated L-group, its flag variety, or Schubert varieties. In the combinatorial theory crystal graphs, Demazure characters, the Schubert calculus and Kazhdan-Lusztig theory all enter.
Recent progress in combinatorial representation theory has been facilitated by the development of computer programs by the Sage-Combinat group within the open-source mathematical software Sage. These tools have evolved in response to the research needs of the developers, but through a disciplined development process are general enough to have already found applications beyond their original intent. A major component of the program will be the further development and application of these resources, and it is hoped that they may be used to investigate the connection to multiple Dirichlet series and related combinatorics.