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A Pieri-Chevalley formula in the K-theory of a $G/B$-bundle

Authors: Harsh Pittie and Arun Ram
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 102-107
MSC (1991): Primary 14M15; Secondary 14C35, 19E08
Published electronically: July 14, 1999
MathSciNet review: 1701888
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Abstract: Let $G$ be a semisimple complex Lie group, $B$ a Borel subgroup, and $T\subseteq B$ a maximal torus of $G$. The projective variety $G/B$ is a generalization of the classical flag variety. The structure sheaves of the Schubert subvarieties form a basis of the K-theory $K(G/B)$ and every character of $T$ gives rise to a line bundle on $G/B$. This note gives a formula for the product of a dominant line bundle and a Schubert class in $K(G/B)$. This result generalizes a formula of Chevalley which computes an analogous product in cohomology. The new formula applies to the relative case, the K-theory of a $G/B$-bundle over a smooth base $X$, and is presented in this generality. In this setting the new formula is a generalization of recent $G=GL_n({\mathbb C})$ results of Fulton and Lascoux.

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Additional Information

Harsh Pittie
Affiliation: Department of Mathematics, Graduate Center, City University of New York, New York, NY 10036

Arun Ram
Affiliation: Department of Mathematics, Princeton University, Princeton, NJ 08544
MR Author ID: 316170

Received by editor(s): February 9, 1999
Published electronically: July 14, 1999
Additional Notes: Research supported in part by National Science Foundation grant DMS-9622985.
Communicated by: Efim Zelmanov
Article copyright: © Copyright 1999 American Mathematical Society