Half the sum of positive roots, the Coxeter element, and a theorem of Kostant
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Dipendra Prasad
Abstract
Interchanging the character and co-character groups of a torus T over a field k introduces a contravariant functor T → T∨. Interpreting ρ : T(ℂ) → ℂ×, half the sum of positive roots for T, a maximal torus in a simply connected semi-simple group G (over ℂ) using this duality, we get a co-character ρ∨ : ℂ× → T∨(ℂ) for which ρ∨(e2πi/h) (h the Coxeter number) is the Coxeter conjugacy class of the dual group G∨(ℂ). This point of view gives a rather transparent proof of a theorem of Kostant on the character values of irreducible finite-dimensional representations of G(ℂ) at the Coxeter conjugacy class: the proof amounting to the fact that in G∨sc(ℂ), the simply connected cover of G∨(ℂ), there is a unique regular conjugacy class whose image in G∨(ℂ) has order h (which is the Coxeter conjugacy class).
© 2016 by De Gruyter
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- Frontmatter
- On Barnes beta distributions, Selberg integral and Riemann xi
- Multi-parameter Triebel–Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition
- Littlewood–Paley functions on homogeneous groups
- Boundedness of singular integrals associated to surfaces of revolution on Triebel–Lizorkin spaces
- The cohomology of the sporadic group J2 over 𝔽3
- Categorical representations of the modular group
- On the Koszul map of Lie algebras
- A twisted Bass–Heller–Swan decomposition for the algebraic K-theory of additive categories
- Stratifications of algebras with two simple modules
- Pro-Lie groups approximable by discrete subgroups
- Half the sum of positive roots, the Coxeter element, and a theorem of Kostant
Articles in the same Issue
- Frontmatter
- On Barnes beta distributions, Selberg integral and Riemann xi
- Multi-parameter Triebel–Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition
- Littlewood–Paley functions on homogeneous groups
- Boundedness of singular integrals associated to surfaces of revolution on Triebel–Lizorkin spaces
- The cohomology of the sporadic group J2 over 𝔽3
- Categorical representations of the modular group
- On the Koszul map of Lie algebras
- A twisted Bass–Heller–Swan decomposition for the algebraic K-theory of additive categories
- Stratifications of algebras with two simple modules
- Pro-Lie groups approximable by discrete subgroups
- Half the sum of positive roots, the Coxeter element, and a theorem of Kostant
