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# Catégories dérivées de blocs à défaut non abélien de GL(2,q)

Abstract : This thesis studies the derived category of the principal block of the finite group GL(2, q) in characteristic l. The theory of Deligne and Lusztig provides two complexes Le and Ls. If l is not equal to 2 then the Sylow l-subgroups of GL(2, q) are abelian. I check that if l divides q-1 (respectively q+1) then the complex Le (respectively Ls) induces a splendid'' derived equivalence between the sum of the blocks of maximal defect of GL(2, q) and the group algebra of the normalizer of a Sylow l-subgroup. This checks Broué's conjecture. If l=2 and q is odd, then the Sylow l-subgroups of GL(2, q) are not abelian. I show that if q is congruent to 1 or 7 modulo 8 then there is no local subgroup H of GL(2, q) such that the principal blocks of H and GL(2, q) have the same type. If q is congruent to 3 or 5, I consider the normalizer in GL(2, q) of a Sylow subgroup of SL(2, q). I show that its principal block has the same type than the principal block of GL(2, q), and that these two blocks are related by a splendid'' derived equivalence. After that I use the theory of A-infinity-algebras. Starting from the complexes Le and Ls, I build a minimal A-infinity-algebra whose derived category is equivalent to the derived category of the principal block of GL(2, q). This construction generalizes the construction of the derived equivalences given in the cases where the Sylow subgroups are abelian. I give a complete description of the A-infinity-algebras obtained when considering PGL(2, q) instead of GL(2, q). In particular I show that the maps m(n) giving the supplementary A-infinity-structure are zero for all n> 3.
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https://tel.archives-ouvertes.fr/tel-00002033
Contributor : Bertrand Gonard Connect in order to contact the contributor
Submitted on : Wednesday, November 27, 2002 - 11:38:51 AM
Last modification on : Friday, March 27, 2020 - 3:33:18 AM
Long-term archiving on: : Friday, April 2, 2010 - 6:40:16 PM

### Identifiers

• HAL Id : tel-00002033, version 1

### Citation

Bertrand Gonard. Catégories dérivées de blocs à défaut non abélien de GL(2,q). Mathématiques [math]. Université Paris-Diderot - Paris VII, 2002. Français. ⟨tel-00002033⟩

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