Representation Theoretical Methods in the Theory of Special Functions

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This project is in two separate major research areas. The generalTheory of m-Quasi-invariants and the Theory of Macdonald Polynomials.m-Quasi-invariants originally arose in the study of particle dynamicsbut have emerged as a truly fascinating Algebraic Combinatorialsubject. For each Coxeter Group G there is an associated one parameterfamily of subspaces which interpolates between the ordinary polynomial ring and the ring of G-invariants. The m-Quasi-Invariants of $G$ are simply polynomials which, upon the action of the difference operator corresponding to a reflecting hyperplane of G, they produce a factor that is divisible by the equation of the reflecting hyperplane raised to the power 2m+1. This theory has been extensively developped, from the algebraic point of view, in fundamental papers by Feigin-Veselov, and Felder-Veselov, and culminated in certain conjectures which were proved by Etingov-Ginsburg in 2002. Jointly with N. Wallach the PI discoveredthat several constructs associated to $m$-quasi-invariants,such as the associated Baker-Achieser function, encode some trulyremarkable combinatorial properties of the corresponding Coxeter group.The present project is to explore the theory from the combinatorialpoint of view. The early successes obtained in the study of the simplest cases of the dihedral groups strongly suggest that such a study should be conducive to significant discoveries in various areas of Mathematics ranging from Combinatorics, Representation Theory and the Theory of special functions.Since its 1988 discovery, the Macdonald symmetric function basis hasprogressively emerged as a central element in the connection betweenRepresentation Theory and the Theory of Symmetric Functions.After more than a decade of research in the Theory of Macdonald polynomials, the PI, M. Haiman and collaborators have been led to a variety of conjectures in Representation Theory, Algebraic Geometry, Combinatorics and Symmetric Function Theory. Efforts in proving these conjectures have yielded fundamental facts and methods in each of these areas. Recently J. Haglund discovered and in collaboration with Haiman and Loehr proved a remarkable purely combinatorial formula for a suitably modified family of Macdonald polynomials. This discovery opens up a variety of research problems involving some of the conjectureswhich up to quite recently appeared inaccesible. The Pi plans to dedidcatea substantial portion of the projected research effort in this area incollaboration with his PhD students. The proposed work, involves extensive computer explorations andtransfer of information across several mathematical boundaries, fromRepresentation Theory to the Theory Special Functions and then to Combinatorics. These transfers provide an invaluable vehicle of discovery, since results and mechanisms which may be quite obvious in one of these areas often translate in highly non trivial and unexpected facts in one of the other areas. It should be mentioned that symbolic manipulation software such as MAPLE and MATHEMATICA combined with the present generation of fast processors have literally transformed many branches of mathematics into experimental sciences. Moreover computer data can be successfully used not only in the discovery of results but also in the construction of proof. The two basic projects that are the focus of the planned research are also highly suitable for the training of young researchers and the opportunity that it provides them to discover and experience the manner in which research can be carried out in the 21st century. Thus all the research problems created by this projectare invariably brought in the the class room to be shared with studentsand collaborators.