Representation theory of finite reductive groups cabanes marc enguehard michel
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The bijection between the sets of simple roots gives rise to a bijectionbetween the sets of parabolic subgroups containing the corresponding Borelsubgroups see Definition 2. The invertibility ofthese basis elements is related to the following quite natural question. I was not aware of this. The methods are a balanced mix ofmodule theory and sheaf theory. Let V be an orthogonal F-space, of even dimension 2n, with a rationalstructure on Fq , of Witt symbol v {0, w}.

Then B is the subgroup of unimodular upper triangular matrices and isthe stabilizer of Fe, U is the stabilizer of e, T is the subgroup of unimodulardiagonal matrices and acts on Fe. This contradicts theirreducibility of W. Hence wFr h has all components equal to 1 apartfrom the component of index h modulo r , which may be written wh t0. The above is true for any C , C bythe basic properties of derived functors. Then , 0 since it is the scalar product of two elements of. The center of H is Fe. Another way of stating that is to define the followingidempotents.

We shall comment onHarish-Chandra induction, cuspidality, Hecke algebras, the Steinberg module,the duality functor, and derived categories. Let M be a simple cuspidal kL I -module. Then it suffices to check that eG F. Indeed, once the duality with p-morphismsbetween G, F and G, F is defined around maximal tori T and T by anisomorphism 8. Let t be the element of W cor-responding to the reflection associated with in the geometric representation.

We may assume that the space V Fq is the space of rational points of anorthogonal F-space V F. Bruhat decomposition If G, B, N , S is a Tits system, the sub-sets BwB wW are distinct and form a partition of G. Let G be a fi-nite group and be a prime. Note that it is enough to check that d1 B generates the same lat-tice as d1 Irr G,b and that B has a cardinality greater than or equalto the expected dimension, i. Once i is proved, this gives ii. Here wh is the word of length h on the r letters s1, s2,. The first gathers the basic knowledge of derivedcategories and derived functors.

Let d0n nN, dwn nN be two sequences of integers. So L above is a set of subquo-tients. Assume that the center of G is connected. From non-connected center to connected centerand dual morphism Hypothesis 15. One hopes that Cambridge will be able to stick to that promise. We first prove a series of propositions about the composition ofthe bn,n M s see Definition 3.

Thus ii follows from i and its proof. Anelementary argument see Proposition 16. To such a class of F-stable Levi subgroups in G therecorresponds a class of F-stable Levi subgroups L in G, whose parame-ter is the coset W I F w. The trivial module for thisring is C with the elements of G acting by IdC. Let F be a locally constant sheaf in Shk Xet. Both imply c the natural map Rc X,F R X,F in Db kmod is an isomor- phism.

I got lost in the next-to-last paragraph of your discussion. All those functors are left-exact. In finite group theory, these are often used as a first step towards thestudy of the full module category, or, more importantly, through the functorsthey define. By Langs theorem Theorem 7. As the form is defined on Fq , the dual of abasis of Fq of one of these spaces i. The second assertion follows from the first by adjunction.

The following will be useful. It will be usedin the rest of the book only through the multiplicity one statement given byTheorem 15. Throughout, the text is illustrated by many examples; background is provided byseveral introductory chapters on basic results, and appendices on algebraic geometryand derived categories. Recall w S {1} r see Definition 10. The integer v is defined so that v 2N if and only if the discriminantof the restriction of the form to V1 is a square in Fq see the beginning of 16. Characters of finite reductive groups -- 9.