51. Lifting of elements of Weyl groups
- Author
-
Jeffrey Adams and Xuhua He
- Subjects
Weyl group ,Algebra and Number Theory ,010308 nuclear & particles physics ,010102 general mathematics ,Tits group ,01 natural sciences ,Combinatorics ,Lift (mathematics) ,20G15 22E10 ,symbols.namesake ,Algebraic group ,Norm (mathematics) ,0103 physical sciences ,symbols ,FOS: Mathematics ,Cartan subgroup ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics - Representation Theory ,Mathematics - Abstract
Suppose G is a reductive algebraic group, T is a Cartan subgroup of G, N = Norm ( T ) , and W = N / T is the Weyl group. If w ∈ W has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.
- Published
- 2016
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