Action of the relative Weyl group on partial Springer sheaf

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 $W(L)$.