2018/06/06 by N. Spaltenstein · 44 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Unipotent #Algebraic group #Injective function #Mathematics #Algebraically closed field #Reductive group #Combinatorics #Conjugacy class #Weyl group #Finite group #Isomorphism (crystallography) #Group (periodic table) #Irreducible representation #Maximal torus #Algebraic closure #Group theory #Physics #Algebraic number #Discrete mathematics #Pure mathematics #Crystallography
paper · pdf · doi:10.2969/aspm/00610317
published in Advanced studies in pure mathematics, 317-338 (Mathematical Society of Japan)
openalex publication_date 2018/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let G be a connected reductive algebraic group defined over an algebraically closed field k.Let Q3G be the variety of Borel subgroups of G, and for x e G let Q3~=B e Q3 G IB;) x.Springer [28] has shown that the Weyl group W of G acts naturally on the .e-adiccohomology groups Hi(Q3~; Qi) (.e a prime, .e*char(k».We shall consider here the action of W defined by Lusztig [13] rather than that defined originally by Springer (they agree up to tensoring by the sign representation of W [10]; moreover no restriction on the characteristic is needed in [13]).If H is a finite group, let H" be the set of all isomorphism classes of irreducible Qrrepresentations of H.For x e G the finite group AG(x)= CaCx)/C~(x) acts on H*(Q3~) (we shall omit the Qt), and this action commutes with that of W. Let d",=dimQ3~.The action of WXAG(x) in H2dz()8~) turns out to be particularly interesting.For any peW" there exist a unipotent element u e G and ¢ e AG(u)" such that p®¢ occurs in H2d"()8~).Moreover the pair (u, ¢) is unique up to G-conjugation, and p®¢ occurs with multiplicity one in H2d"()8~).We write then p=p~,~.This injective map from W" to the set JIf G of conjugacy classes of pairs (u, ¢) (u e G unipotent, ¢ e AG(u)") is explicitly known in most cases.If the characteristic is good, it is described by Shoji [22] [23] (G classical, F4), Springer [28] (Gz), Alvis, Lusztig and the author [3] (En. n=6, 7, 8).Classical groups in characteristic 2 are treated in [17].We shall consider here exceptional groups in bad characteristic.We shall also be concerned with Lusztig's generalization of Springer's correspondenceLusztig attaches to (u, ¢) a 4-tuple (L, v, t, p) (or rather a G-conjugacy class of such objects), where L is a Levi factor of some parabolic subgroup of G, veL is unipotent, t e Aiv)", p e (NG(L)/L)".If L=G, then (v, t) is conjugate to (u, ¢) and p is automatically trivial, and we say that (u, ¢) is cuspidal for G. Lusztig's construction has in particular the following properties.It defines a bijection between JIf G and the set fll' G of G-conjugacy classes of 4-tuples (L, v, t, p) as above for which (v, t) is cuspidal