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On the irreducibility of irreducible characters of simple Lie algebras


Author: C. S. Rajan
Journal: Trans. Amer. Math. Soc. 366 (2014), 6443-6481
MSC (2010): Primary 17B10; Secondary 20G05
Published electronically: July 25, 2014
MathSciNet review: 3267015
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Abstract: We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors.

For $ SL(r)$ the irreducibility result is the following: let $ \lambda =(a_1\geq a_2\geq \cdots \geq a_{r-1}\geq 0)$ be the highest weight of an irreducible rational representation $ V_{\lambda }$ of $ SL(r)$. Assume that the integers $ a_1+r-1, ~a_2+r-2, \cdots , a_{r-1}+1$ are relatively prime. Then the character $ \chi _{\lambda }$ of $ V_{\lambda }$ is strongly irreducible in the following sense: for any natural number $ d$, the function $ \chi _{\lambda }(g^d), ~g\in SL(r,\mathbb{C})$ is irreducible in the ring of regular functions of $ SL(r,\mathbb{C})$.


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C. S. Rajan
Affiliation: Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay, 400 005, India
Email: rajan@math.tifr.res.in

DOI: https://doi.org/10.1090/S0002-9947-2014-06080-7
Received by editor(s): August 17, 2012
Received by editor(s) in revised form: January 15, 2013
Published electronically: July 25, 2014
Article copyright: © Copyright 2014 American Mathematical Society