Commutation of Shintani descent and Jordan decomposition

Let GF be a finite group of Lie type, where G is reductive defined over F¯q and F Frobenius root. Lusztig’s Jordan decomposition parametrises the irreducible characters in rational series E(GF,(s)G∗F∗) s∈G∗F∗ by E(CG∗(s)F∗,1). We conjecture that Shintani twisting preserves space class functions generated union E(GF,(s′)G∗F∗) (s′)G∗F∗ runs semi-simple classes G∗F∗ geometrically conjugate to s; further, extending linearity this space, we there way fix such it maps on disconnected groups Deshpande, which acts linear span ∐s′E(CG∗(s′)F∗,1). show non-trivial case conjecture, type An−1 with n prime.