Stabilisers of eigenvectors in complex reflection groups

Abstract

Eigenvectors of elements of real and complex reflection groups have been studied byCoxeter, Kostant, Springer, Lehrer and Bessis amongst others, due to their rich geometry, and the extensive information that they yield about the structure of the reflection group and related objects in Lie theory and braid theory. Recently, Kamgarpour proved that if G is an irreducible finite real reflection group of rank n, x is an eigenvector of any element of G with eigenvalue a primitive dth root of unity, and Φ and Φx denote the root systems of G and StabG(x) respectively, then |Φ| − |Φx| ≥ dn, with equality if and only if d is the Coxeter number.In this thesis, we prove the following generalisation of Kamgarpour’s inequality. If G is an irreducible complex reflection group of rank n and x is any eigenvector of an element of G with eigenvalue a primitive dth root of unity, then we havel(π ) − l(π x) ≥ dn,with equality if and only if G is well-generated and d = h. Here, π is the generator of the centre of the pure braid group of G, π x is the generator of the centre of the pure braid group of the stabiliser of x, and l denotes the length function on the braid group of G. Our proof is case-by-case using the Shephard–Todd classification of complex reflection groups. We also investigate the case where l(π ) − l(π x) − dn is as small as possible while still being positive, as well as a possible braid-theoretic explanation of our result.