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T-equivariant K-theory of generalized flag varieties
Summary
We introduce a new ring construction for Kac-Moody groups, yielding T-equivariant K-theory and Demazure-like operators. This framework unifies structures in algebraic geometry and representation theory, with potential new results for finite-dimensional groups.
Area of Science:
- Algebraic Geometry
- Representation Theory
- K-theory
Background:
- Kac-Moody groups and their associated structures like Borel subgroups and maximal tori are central to modern mathematics.
- Previous work by Kostant and Kumar established foundational connections between Weyl groups and algebraic structures.
Purpose of the Study:
- To define a novel ring Y based on the Weyl group and its action on the maximal torus of a Kac-Moody group.
- To establish a canonical isomorphism between the dual of ring Y (ring Psi) and the T-equivariant K-theory of the flag variety G/B.
- To demonstrate that the structures on K(T)(G/B), including operators analogous to Demazure operators, arise naturally from ring Y.
Main Methods:
- Construction of ring Y using the Weyl group W and its action on the maximal torus T.
- Dualization of ring Y to obtain ring Psi.
- Establishing a canonical isomorphism between Psi and K(T)(G/B).
- Analysis of operators on K(T)(G/B) and their connection to ring Y.
Main Results:
- A new ring Y is defined based on Weyl group actions.
- Ring Psi is shown to be canonically isomorphic to the T-equivariant K-theory K(T)(G/B).
- The structures on K(T)(G/B), including Demazure-like operators, are shown to originate from ring Y.
- Evaluation of Psi at 1 recovers the K-theory K(G/B) with its structures.
Conclusions:
- The developed framework provides a unified approach to understanding K-theory and related structures in the context of Kac-Moody groups.
- The results offer new insights and potentially novel findings, particularly for the finite-dimensional semisimple group case.
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