Kazhdan-Lusztig conjecture

introduction

• The Kazhdan-Lusztig theory provides the solution to the problem of determining the irreducible characters in the BGG category \(\mathcal{O}\) of semisimple Lie algebras ([KL], [BB], [BK]).
• The theory was originally formulated in terms of the canonical bases (i.e., Kazhdan-Lusztig bases) of Hecke algebras.
• 1979 conjectures
  • KL character formula
  • KL positivity conjecture
• Kazhdan-Lusztig polynomial

Hecke algebra

• basis of Hecke algebra \(\{H_{x}| x\in W\}\)
• new basis of Hecke algebra \(\{\underline{H}_{x}| x\in W\}\)

\[ \underline{H}_{x}=H_{x}+\sum_{y\in W, \ell(y)<\ell(x)} h_{y,x}H_{y} \] where \(h_{y,x}\in v\mathbb{Z}[v]\) is so called the Kazhdan-Lusztig polynomial

• positivity conjecture \[h_{x,y}\in \mathbb{Z}_{\geq 0}[v]\]

Hodge theory

• Poincare duality
• hard Lefshetz theorem
• Hodge-Riemann bilinear relation

related items

• BGG category
• Hecke algebra
• Enumerative problems and Schubert calculus
• Flag manifold and flag variety

exposition

• Williamson- Kazhdan-Lusztig conjecture and shadows of Hodge theory

articles

• [BB] A. Beilinson and J. Bernstein, Localisation de \(\mathfrak g\)-modules, C.R. Acad. Sci. Paris Ser. I Math. 292 (1981), 15-18.
• [BK] J.L.Brylinski and M.Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410.
• [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.

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• ID : Q6381065

Spacy 패턴 목록

• [{'LOWER': 'kazhdan'}, {'OP': '*'}, {'LOWER': 'lusztig'}, {'LEMMA': 'polynomial'}]