Fourier Transforms of Invariant Functions on Finite by Emmanuel Letellier PDF
By Emmanuel Letellier
The research of Fourier transforms of invariant services on finite reductive Lie algebras has been initiated by way of T.A. Springer (1976) in reference to the geometry of nilpotent orbits. during this booklet the writer reports Fourier transforms utilizing Deligne-Lusztig induction and the Lie algebra model of Lusztig’s personality sheaves conception. He conjectures a commutation formulation among Deligne-Lusztig induction and Fourier transforms that he proves in lots of situations. As an software the computation of the values of the trigonometric sums (on reductive Lie algebras) is proven to minimize to the computation of the generalized eco-friendly capabilities and to the computation of a few fourth roots of unity.
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Additional resources for Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras
F (φ) ◦ φ. 9. Let (K, φ) and (K , φ ) be two F -equivariant semi-simple perverse sheaves on X such that for any n ≥ 1, we have XK,φ(n) = XK ,φ (n) . Then K and K are isomorphic in M(X). 10. Let H be a connected linear algebraic group deﬁned over Fq acting morphically on X. We assume that this action is deﬁned over Fq and we still denote by F : H → H the Frobenius on H. 11. Let K be an H-equivariant F -stable complex (or sheaf) on X and let φ : F ∗ (K) → K be an isomorphism. e. x) = XK,φ (x). 16, is F -stable if Z and E are both F -stable.
28. 27(1), we write RGL instead of RGL⊂P . 29. We now state our conjecture about a commutation formula between Fourier transforms and Deligne-Lusztig induction. 9. 30. For any F -stable Levi subgroup L of G and any function f ∈ C(LF ), we have F G ◦ RGL (f ) = where G G G L RL ◦ F L (f ) = (−1)Fq −rank(G) . 11(i)). 30 will be discussed in chapter 6. 4 Local Systems and Perverse Sheaves In this chapter we introduce the results on local systems and perverse sheaves which will be used in the following next chapters.
4. Assume that P , L and K are all F -stable and let φ : F ∗ (K) → K be an isomorphism. 2). Indeed, if we denote by F2 the Frobenius on V2 deﬁned by F2 (x, hP ) = (F (x), F (h)P ) and by ∼ ˜ ˜ → K the isomorphism induced by φ, then (*) follows easily from φ˜ : F2∗ (K) the formula Xπ (K),ψ (y) = XK, ˜ ˜ φ ˜ (x) ! x∈(π −1 (y))F2 which is a consequence of the Grothendieck trace formula. 5. We have an isomorphism of functors indGL⊂P ◦ DL DG ◦ indGL⊂P . Proof: Since the morphisms π and π are smooth with connected ﬁbers of same dimension and since the morphism π is proper, we get the following relations: (i) DV1 ◦ (π ∗ [m]) π ∗ [m] ◦ DL , (ii) DV1 ◦ ((π )∗ [dim P ]) (π )∗ [dim P ] ◦ DV2 , (iii) DG ◦ (π )!
Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras by Emmanuel Letellier