Lifting involutions in a Weyl group to the normalizer of the torus

Abstract

Let N N be the normalizer of a maximal torus T T in a split reductive group over F q \mathbb {F}_q and let w w be an involution in the Weyl group N / T N/T . We construct a section of W W satisfying the braid relations, such that the image of the lift n n of w w under the Frobenius map is equal to the inverse of n n .