  In the context of the Springer correspondence, the Weyl group action on the
Springer sheaf can be defined in two ways: via restriction or the Fourier
transform. It is well-known that these two actions differ by the sign
character. This was proven by Hotta for sheaves with characteristic 0
coefficients in 1981, and more recently extended to arbitrary coefficients by
Achar, Henderson, Juteau, and Riche in 2014. In this short article, we study an
extension of this problem to the partial Springer sheaf with arbitrary field
coefficients. This involves an action of the so-called relative Weyl group
$W(L)$.
