The Weyl vector rho for a chosen positive system

Depends on

• Triangular decomposition from a chosen positive root system

Used by

• Kostant cohomology in degrees zero and top Corollary
• Tensor product with a minuscule representation Corollary
• The Kostant Euler character recovers the Weyl numerator Corollary
• Omitting the exterior root-weight shifts gives the wrong dot weight Counterexample
• Omitting the rho shift breaks Kostant's formula Counterexample
• The BGG complex cannot be used unchanged at a singular weight Counterexample
• Dot-Weyl facets and single-wall translation data Definition
• Flag-variety twisted differential operators and their modules Definition
• The Bruhat graph and the BGG Verma sum in degree k Definition
• The classical BGG category O Definition
• The integral Weyl group of a weight Definition
• The inversion set of a Weyl group element Definition
• The Jantzen deformation and filtration of a Verma module Definition
• The strong linkage order on weights Definition
• Freudenthal recursion for the sl3 adjoint zero weight Example
• Kostant cohomology for sl2 Example
• Kostant cohomology for the trivial sl3 module Example
• Kostant multiplicity in the sl3 adjoint module Example
• The A2 Weyl denominator expansion Example
• The BGG resolution for sl2 Example
• The Clebsch--Gordan tensor decomposition for sl2 Example
• The zero-weight singular central character Example
• Weyl character and dimension formulas for sl2 Example
• Dominant integral weights are maxima of their Weyl orbits Lemma
• Finite-dimensional tensoring reaches every simple of a linkage class Lemma
• Partial-flag twist and local product decomposition Lemma
• Positive coroot pairings of a dominant integral weight Lemma
• Preliminary factorization of a Shapovalov determinant Lemma
• Regularized evaluation of the Weyl character quotient at one Lemma
• The augmentation kernel is the sum of the simple-reflection Verma submodules Lemma
• The BGG Euler identity gives the Weyl numerator Lemma
• The Casimir comparison on a weight space Lemma
• The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
• The difference of the Weyl vector from its reflections is a sum of positive roots Lemma
• The extremal weight cochain of a Weyl element is closed and unique Lemma
• The rho-shift intertwines the dot and ordinary Weyl actions Lemma
• The shifted norm of a weight is maximal only at the top weight Lemma
• The simple-root singular vector in a Verma module Lemma
• The singular simple-parabolic kernel on the flag variety Lemma
• Weight subsets with equal root sums are unique Lemma

…and 6 more results.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Yiannis Sakellaridis, Verma Modules and the Category O (standard reference, not scraped)