RUI: Geometry of Conjugacy and K-Theory in Affine Weyl Groups

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We encounter symmetry in nearly every aspect of our daily lives: looking at our faces in the mirror, watching snowflakes fall from the sky, and driving across bridges. Symmetric organisms persist through evolution in nature, symmetric protagonists are perceived as especially beautiful in art, and symmetric components are critical to engineering structures that can withstand powerful forces. The set of symmetries of a particular physical object enjoys a rich algebraic structure, because symmetries are operations that can be composed together. This group of all symmetries can then be conveniently studied by encoding each symmetry as a rectangular array of numbers called a matrix. This process of passing from a symmetric object in the natural world to a related collection of matrices is the hallmark of the mathematical field of representation theory. Representation theory thus reduces the complex study of symmetry in nature to questions in the well understood area of mathematics called linear algebra. As such, the proposed projects have broad potential to substantially impact our understanding of many symmetric structures occurring throughout the mathematical and natural sciences. This project also provides opportunities for directly involving undergraduate students in mathematical research, with a focused goal of supporting the development and recruitment of women in mathematics.This project will address two topics in the algebra, geometry, and representation theory of reductive algebraic groups over non-archimedean local fields. First, applying techniques from geometric group theory to the associated Bruhat-Tits building, the investigator will provide a global approach to understanding the conjugacy classes of any affine Weyl group. Second, this geometric perspective will also be applied to reinterpret the K-theoretic generalization of Peterson's isomorphism from the equivariant homology of the affine Grassmannian to the equivariant quantum cohomology of a finite flag variety. Specific objectives include a complete description of an affine conjugacy class in terms of the underlying finite Weyl group, and new representatives for the Schubert classes in the equivariant K-homology of the affine Grassmannian. As such, the research will stimulate new interactions among the mathematical subfields of representation theory, arithmetic geometry, enumerative geometry, algebraic combinatorics, geometric group theory, and mathematical physics.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.