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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.

Copyright 1999 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 05 (1999), pp. 102-107
• Publisher Identifier: S 1079-6762(99)00067-0
• 1991 Mathematics Subject Classification. Primary 14M15; Secondary 14C35, 19E08
• Key words and phrases.
• Received by the editors February 9, 1999
• Posted on July 14, 1999
• Communicated by Efim Zelmanov
• Comments (When Available)

Harsh Pittie

Department of Mathematics, Graduate Center, City University of New York, New York, NY 10036

Arun Ram

Department of Mathematics, Princeton University, Princeton, NJ 08544

E-mail address: rama@math.princeton.edu

Research supported in part by National Science Foundation grant DMS-9622985.