By James Dugundji

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We denote by MH (X) the subcategory of M(X) consisting of H-equivariant perverse sheaves and whose morphisms are deﬁned in the same way as we deﬁned morphisms in ShH (X). As we did for sheaves, we can prove that MH (X) is in fact a full subcategory of M(X). 5 works also for perverse sheaves. 11. 8 on H-equivariance is not the appropriate one for the case where K ∈ Dcb (X) is not a perverse sheaf or when H is not connected. For the general deﬁnition of H-equivariance see [BL94]. 12. 34, the Verdier dual of an H-equivariant perverse sheaf on X is H-equivariant, hence the restriction of DX to MH (X) is an equivalence of categories MH (X) → MH (X).

Let ˜j : H × X → H × H × X, (g, x) → (1, g, x) so that j = ˜j ◦ i and (α × IdX ) ◦ ˜j = IdH×X . Let H acts on H × X by left multiplication on the ﬁrst coordinate. 3 we get that ˜j ∗ (α × IdX )∗ (φE ) = ˜j ∗ (IdH × ρ)∗ (φE ) ◦ (p2 × IdX )∗ (φE ) . 3, we get that ˜j ∗ is an equivalence of categories from the full subcategory of Sh(H × H × X) of H-equivariant sheaves on H × H × X (H acting by left multiplication on the ﬁrst coordinate) onto Sh(H × X). Hence it remains to see that (α × IdX )∗ (φE ) is a morphism between H-equivariant sheaves.

We now state the transitivity property of Deligne-Lusztig induction. Let M ⊂ L be an inclusion of F -stable Levi subgroups of G with respective Lie algebras M and L. Let P and Q be two parabolic subgroups of G, having respectively L and M as Levi subgroups, such that Q ⊂ P . 22. We have RGL⊂P ◦ RL M⊂L∩Q = RM⊂Q . We have the following proposition. 23. If the parabolic subgroup P is F -stable, then the DeligneLusztig induction RGL⊂P coincides with Harish-Chandra induction. 24. Let L be an F -stable Levi subgroup of G and P be a parabolic subgroup of G having L as a Levi subgroup.