Universal modules in natural characteristic for finite reductive groups.
Let \(k\) be a finite field with characteristic denoted by \(p\), let \(\mathbf G\) be a connected reductive algebraic group defined over \(k\), and let \(\mathbf U\) be the unipotent radical of a Borel subgroup \(\mathbf B\) defined over \(k\). Let \(G\), \(U\) and \(B\) be the \(k\)-rational points of \(\mathbf G\), \(\mathbf U\) and \(\mathbf B\), respectively. Define \(C\) as the space of functions on \(G\) with values in an algebraic closure \(\overline k\) of \(k\) that are invariant by left translations by \(U\). Then \(C\) is called the \textit{universal module} for \(G\). The group \(G\) acts by right translation on \(C\). The commuting algebra \(H\) of the resulting representation of \(G\) is the Hecke algebra of \(G\) with respect to \(U\). The aim of the article is the study of the representations of \(G\) with coefficients in \(\overline k\), and the links of the latters with \(H\)-modules. It is motivated by (and used in) the study of mod \(p\) representations of \(p\)-adic reductive groups. Since \(C\) is both a representation of \(G\) and a left \(H\)-module, there is a functor \(\mathbf F\colon V\mapsto\Hom_G(C,V)\) from the category of \(\overline k\)-representations of \(G\) to the category of right \(H\)-modules. In general this functor is not exact, because the order of \(U\) is not invertible in \(\overline k\). Nevertheless, it induces a bijection between the isomorphism classes of irreducible representations of \(G\) and those of simple \(H\)-modules. It admits as right adjoint the functor \(E\mapsto E\otimes_C H\). The first main result proved in the article is the equivalence of the flatness of the \(H\)-module \(C\) with the fact that the functor \(\mathbf F\) induces an equivalence between the category of finite dimensional \(\overline k\)-representations of \(G\) that are generated by their \(U\)-invariant vectors and the category of finite dimensional right \(H\)-modules. This result is part of Proposition 2.5, which provides also several other criteria for the flatness of \(C\). In the case when \(\mathbf G=\mathrm{GL}_n\), \(n\geq 1\), the authors prove much more precise results. Let \(C_1\) denote the space of functions on \(G\) with values in \(\overline k\) that are invariant by left translations by \(B\), and let \(H_1\) be the commuting algebra of \(C_1\). Let \(q\) denote the order of \(k\). Then the following holds (see Proposition 5.16): (1) If \(n=2\) then the \(H_1\)-module \(C_1\) is flat. (2) If \(n=2\) then the \(H\)-module \(C\) is flat if and only \(q=p\). (3) If \(n=3\) then the \(H_1\)-module \(C_1\) is flat if and only if \(q=p\). (4) If \(n=3\) then the \(H_1\)-module \(C\) is flat if and only if \(q=2\). (5) If \(n\geq 4\) and \(q\neq p\), then the \(H_1\)-module \(C_1\) is not flat. (6) If \(n\geq 4\) and \(q\neq 2\), then the \(H\)-module \(C\) is not flat.
scientific article; zbMATH DE number 6496545
scientific article