Complete reducibility of quadratic modules for finite Lie-type groups. (Q713283)

Let \(G\) be a Lie-type group over a finite field of odd characteristic \(p\) and \(V\) be a \(\mathbb Z_pG\)-module, on which any root group \(A\) of \(G\) acts quadratically, i.e. \([V,A,A]=0\) (if \(V\) is nontrivial, the root groups of \(G\) must be Abelian). In the present paper, it is proved that if \(G\neq\mathrm{SL}_2(3)\) then \(V\) is completely reducible. The author does not use the classification of the irreducible quadratic modules for \(G\) obtained by \textit{A. A. Premet} and \textit{I. D. Suprunenko} [Math. Nachr. 110, 65-96 (1983; Zbl 0522.20027)].