NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Initial Amendment Date:	April 7, 2022
Latest Amendment Date:	April 7, 2022
Award Number:	2153741
Award Instrument:	Standard Grant
Program Manager:	James Matthew Douglass mdouglas@nsf.gov  (703)292-2467 DMS  Division Of Mathematical Sciences MPS  Directorate for Mathematical and Physical Sciences
Start Date:	July 1, 2022
End Date:	June 30, 2025 (Estimated)
Total Intended Award Amount:	$207,196.00
Total Awarded Amount to Date:	$207,196.00
Funds Obligated to Date:	FY 2022 = $207,196.00
History of Investigator:	George Lusztig (Principal Investigator) GYURI@MATH.MIT.EDU
Recipient Sponsored Research Office:	Massachusetts Institute of Technology 77 MASSACHUSETTS AVE CAMBRIDGE MA  US  02139-4301 (617)253-1000
Sponsor Congressional District:	07
Primary Place of Performance:	Massachusetts Institute of Technology 77 Massachusetts Ave. Cambridge MA  US  02139-4301
Primary Place of Performance Congressional District:	07
Unique Entity Identifier (UEI):	E2NYLCDML6V1
Parent UEI:	E2NYLCDML6V1
NSF Program(s):	ALGEBRA,NUMBER THEORY,AND COM
Primary Program Source:	01002223DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):
Program Element Code(s):	126400
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures, using groups of invertible matrices. Representation theory of finite groups has numerous applications to other areas, including number theory and mathematical physics. In this project the linear structures are themselves finite matrix groups, or more generally matrix groups whose entries satisfy divisibility properties with respect to a fixed prime number. Geometric and combinatorial techniques will be brought to bear to study representations of these groups, especially in the important case when the representing matrices themselves have entries in a finite field.

More precisely, the central aims of this project are to (1) investigate a new approach to representations of Weyl groups and unipotent representations of finite reductive groupsin terms of a new basis of the Grothendieck group; (2) investigate a new formulation of the character formula for semisimple groups in positive characteristic; (3) study Hecke algebras with unequal parameters in the framework of the theory of parabolic character sheaves; (4) study strata of a reductive group; and finally (5) investigate new W-graphs associated to involutions in two-sided cells of a Coxeter groups.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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