A (-q)-analogue of weight multiplicities

2012/03/02 by G. Lusztig, George Lusztig, Lusztig, George +2
Chemistry · Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Molecular spectroscopy and chirality #Primary 20G05 #Representation Theory (math.RT) #Secondary 14D24 #math.RT #msc:14D24 #msc:20G05

paper · pdf · doi:10.48550/arxiv.1203.0521

21 pages

arxiv created 2012/03/02 · openalex publication_date 2012/03/02 · arxiv updated 2012/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We prove a conjecture in \citeL stating that certain polynomials P^σ_y,w(q) introduced in \citeLV1 for twisted involutions in an affine Weyl group give (-q)-analogues of weight multiplicities of the Langlands dual group \checkG. We also prove that the signature of a naturally defined hermitian form on each irreducible representation of \checkG can be expressed in terms of these polynomials P^σ_y,w(q).

Citations

• Hecke algebras and involutions in Weyl groups
• A bar operator for involutions in a Coxeter group

Related

• Parabolic restrictions and double deformations of weight multiplicities
• On Lusztig's q-analogues of all weight multiplicities of a representation
• Weight multiplicities and Young tableaux through affine crystals
• Computing weight q-multiplicities for the representations of the\n simple Lie algebras
• Weight q-multiplicities for representations of the exceptional Lie algebra \mathfrakg₂