## On unipotent blocks and their ordinary characters.(English)Zbl 0817.20046

Let $$G$$ be a connected, reductive algebraic group defined over $$F_ q$$, $$F : G \to G$$ a Frobenius morphism and $$G^ F$$ the finite group of $$F$$- fixed points of $$G$$. Let $$\ell$$ be an odd prime not dividing $$q$$. Assuming that $$\ell$$ is a good prime for $$G$$ and $$\ell \neq 3$$ if $$^ 3 D_ 4$$ is involved in $$G$$, the authors describe the unipotent $$\ell$$- blocks of $$G$$, i.e. the $$\ell$$-blocks containing unipotent characters. Let $$L$$ be an $$F$$-stable Levi subgroup of $$G$$ and $$d$$ a positive integer. Then $$L$$ is said to be $$d$$-split if it is the centralizer of a $$\phi_ d$$-subgroup of $$G$$ in the sense of M. Broué and G. Malle [Math. Ann. 292, 241-262 (1992)], where $$\phi_ d(x)$$ is the $$d$$th cyclotomic polynomial. Let $$R^ G_ L$$ and $$^*R^ G_ L$$ denote the Lusztig functor and its adjoint. A unipotent $$d$$-pair is a pair $$(L,\lambda)$$ where $$L$$ is a $$d$$-split Levi subgroup of $$G$$ and $$\lambda$$ is a unipotent character of $$L^ F$$, and $$(L,\lambda)$$ is $$d$$-cuspidal if for every proper $$d$$-split Levi subgroup $$M$$ of $$L$$, $$^*R^ L_ M(\lambda) = 0$$.
The main theorem (4.4) proved by the authors then states that there is a bijection between $$G^ F$$-conjugacy classes of unipotent $$e$$-cuspidal pairs, where $$e$$ is the order of $$q \bmod \ell$$, and the set of unipotent $$\ell$$-blocks of $$G^ F$$. If a block corresponds to a pair $$(L,\lambda)$$, the unipotent characters in the block are precisely the constituents of $$R^ G_ L(\lambda)$$. The defect group of the block is a Sylow $$\ell$$-subgroup of $$C^ 0_ G([L,L])^ F$$. Furthermore, all the characters in the block can be described in terms of Lusztig maps $$R^ G_{G(t)}$$ where $$t$$ is an $$\ell$$-element in the dual group $$G^{*F}$$, and $$G(t) \subset G$$ is in duality with $$C^ 0_{G*}(t)$$. The proof involves an analysis of the centralizers of $$\ell$$-subgroups and $$e$$-split Levi subgroups, and local block theory.
Remark: The theorem was known for special classes of groups, but the authors remark that their proof is more intrinsic. For large $$\ell$$, when the defect group is abelian, the general theorem was also proved by M. Broué, G. Malle and J. Michel [Astérisque 212, 7-92 (1993)].

### MSC:

 20G05 Representation theory for linear algebraic groups 20C20 Modular representations and characters 20G40 Linear algebraic groups over finite fields
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### References:

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