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The integral Weyl group of a weight
Definition
Fix a finite-dimensional complex semisimple Lie algebra , a Cartan subalgebra , and a positive Borel . Write , when , and .
For a weight , define
Reflections and coroots are those of Root reflections and the Weyl group action, with the shift from The Weyl vector rho for a chosen positive system. The integral-reflection linkage class through is . The word integral includes zero and negative integral pairings. If is empty the generated group is . This definition uses generating reflections; no identification with a root-lattice-coset stabilizer is assumed.
Depends on
Used by
- Dot-Weyl facets and single-wall translation data Definition
- The Bruhat graph and the BGG Verma sum in degree k Definition
- The regular integral block and its antidominant weight Definition
- The singular integral block and its dot stabilizer Definition
- Truncation at a finite downward-closed ideal of a linkage class Definition
- A singular integral A2 central-character summand Example
- Nonintegral sl2 central characters split into two simple blocks Example
- Dominant integral dot translates embed canonically in the Verma module Lemma
- Dominant integral weights are maxima of their Weyl orbits Lemma
- Finite-dimensional tensoring reaches every simple of a linkage class Lemma
- Integral reflection linkage is an equivalence relation Lemma
- The regular coset-dominant Harish–Chandra tensor comparison Lemma
- Central-character summands refine into linkage blocks Theorem
Dependency tree · two levels
5 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
- Humphreys, §3.4 definitions and invariance identities, p.52; §4.9, p.83 (reflection-generated convention) (standard reference, not scraped)