On certain projective modules for finite groups of Lie type (Q2639955)

Let K be the algebraic closure of the finite field \({\mathbb{F}}_ q\). Let \(G_ 0\) be a classical linear subgroup of GL(n,q), V its standard module and St the Steinberg module. In [The discrete series of \(GL_ n\) over a finite field (Ann. Math. Stud. 81, 1974; Zbl 0293.20038)] \textit{G. Lusztig} has shown that for \(q>2\) and \(G_ 0=GL(n,q)\) the projective module \(V\otimes St\) is indecomposable. The author deals with the groups \(SL(l+1,q)\), Sp(2l,q), \(\Omega (2l+1,q)\), \(\Omega \pm (2l,q)\) and \(SU(l+1,q)\) where \(q=2\) is included. The module \(V\otimes St\) is again indecomposable unless one of the following cases holds. \(\Omega (2l+1,q)\) (q\(\geq 3)\), \(SL(l+1,2)\), \(Sp(2l,2)\cong \Omega (2l+1,2)\), \(\Omega \pm (2l,2)\). In the exceptional situations \(V\otimes St\) is the sum of St and a projective indecomposable module. The non-Steinberg part is always described by the weight. For \(SL(l+1,q)\) and \(SU(l+1,q)\) even all the modules \(St\otimes \bigwedge^ kV\) (k\(\leq l)\) are indecomposable whenever \(q\geq 3\). For \(q=2\) the splitting is given.