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

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



.