NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	UNIVERSITY OF NORTH CAROLINA AT CHAPEL HILL
Initial Amendment Date:	April 25, 2002
Latest Amendment Date:	February 20, 2004
Award Number:	0200276
Award Instrument:	Continuing Grant
Program Manager:	Tomek Bartoszynski tbartosz@nsf.gov  (703)292-4885 DMS  Division Of Mathematical Sciences MPS  Directorate for Mathematical and Physical Sciences
Start Date:	May 1, 2002
End Date:	April 30, 2006 (Estimated)
Total Intended Award Amount:	$111,483.00
Total Awarded Amount to Date:	$111,483.00
Funds Obligated to Date:	FY 2002 = $37,161.00 FY 2003 = $37,161.00 FY 2004 = $37,161.00
History of Investigator:	Ivan Cherednik (Principal Investigator) chered@math.unc.edu
Recipient Sponsored Research Office:	University of North Carolina at Chapel Hill 104 AIRPORT DR STE 2200 CHAPEL HILL NC  US  27599-5023 (919)966-3411
Sponsor Congressional District:	04
Primary Place of Performance:	University of North Carolina at Chapel Hill 104 AIRPORT DR STE 2200 CHAPEL HILL NC  US  27599-5023
Primary Place of Performance Congressional District:	04
Unique Entity Identifier (UEI):	D3LHU66KBLD5
Parent UEI:	D3LHU66KBLD5
NSF Program(s):	ALGEBRA,NUMBER THEORY,AND COM
Primary Program Source:	app-0103  app-0104  app-0102
Program Reference Code(s):	0000, OTHR
Program Element Code(s):	126400
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

The purpose of this project is to study double affine Hecke algebras (DAHA), as previously introduced by the principal investigator. They have proved to be very useful in representation theory, the theory of special and spherical functions, conformal field theory, and combinatorics. The main objectives include: (1) A general classification of the semisimple, spherical, unitary, and Fourier-univariant representations of DAHA; (2) the classification of finite dimensional representations of DAHA for generic q, with applications to Macdonald's eta-type identities, Kac-Moody characters, and degenerate Bessel functions; (3) the theory of representations of DAHA at roots of unity, connections with the monodromy of the double affine KZ equations and elliptic braid groups; (4) classificaton of the Fourier-invariant uitary representations of DAHA with applications to Gauss-Selberg sums and Verlinde algebras; (5) harmonic analysis on semisimple representations of DAHA in the compact and noncompact cases, at roots of unity, and in the elliptic case.

This is a project in the area of mathematics known as Representation Theory. The nature of representation theory, and what makes it so useful in so many areas of mathematics and science, is that it is a way of studying and encoding symmetries, whereever they may occur, in nature or abstractly. The purpose of this project is to study and develop the properties of a new representation-theoretic tool, "double affine Hecke algebras," that can be used in the application of representation theory to physics (conformal field theory) and combinatorics. Combinatorics is the science of counting the possible arrangements and ways of organizing collections of anything (e.g., atoms, pebbles, star clusters).

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