Root Multiplicities of the Indefinite Kac–Moody Algebras of Symplectic Type

Abstract

We study the root multiplicities of the indefinite Kac–Moody algebras by viewing them as weight multiplicities of certain integrable -modules. Then using Kang's root multiplicity formula and the path crystal for integrable -modules we calculate the multiplicities of a family of roots for . In particular, we show that for any positive integer k the multiplicity of − 2α₋₁ − kδ as a root of is a polynomial in n of degree at most k. Furthermore, we observe that Frenkel's conjectured root multiplicity bound does not hold for roots of .

Key Words:

• Crystal bases
• Kac–Moody algebras
• Root multiplicities

2000 Mathematics Subject Classification:

• 17B67
• 17B10

ACKNOWLEDGMENT

This article is partially supported by NSA grant MDA904-02-1-0072.

Notes

Communicated by J. Kuzmanovich.