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Principal blocks and the Steinberg character. (English) Zbl 1194.20012
From the text: We determine the finite simple groups of Lie type of characteristic $$p$$, for which the Steinberg character lies in the principal $$\ell$$-block for every prime $$\ell\neq p$$ dividing the order of the group.
Theorem. Let $$G$$ be a finite simple group of Lie type of characteristic $$p$$. Then the Steinberg character of $$G$$ lies in the principal $$\ell$$-block of $$G$$ for all primes $$\ell\neq p$$ dividing the order of $$G$$, if and only if $$G$$ is one of the groups in the following list:
(i) $$\text{PSL}_n(q)$$ with $$2\leq n\leq 4$$ and $$(n,q)\neq(2,2),(2,3)$$.
(ii) $$\text{PSU}_n(q)$$ with $$3\leq n\leq 4$$ and $$(n,q)\neq(3,2)$$.
(iii) $$\text{PSp}_4(q)$$ with $$q\neq 2$$.
(iv) $$\text{P}\Omega^+_8(q)$$.
(v) $$G_2(q)$$ with $$q\neq 2$$.
(vi) $$F_4(q)$$.
(vii) $$^3D_4(q)$$.
(viii) $$^2B_2(q)$$ with $$q=2^{2m+1}>2$$.
(ix) $$^2G_2(q)$$ with $$q=3^{2m+1}>3$$.
(x) $$^2F_4(q)$$ with $$q=2^{2m+1}>2$$.

##### MSC:
 20C33 Representations of finite groups of Lie type 20C15 Ordinary representations and characters
GAP; CHEVIE
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