Kostant proved a celebrated result which computes the ordinary Lie algebra cohomology for the nilradical of the Borel subalgebra of a complex simple Lie algebra \(\mathfrak g\) with coefficients in a finite-dimensional simple \(\mathfrak g\)-module. Later, other proofs have been discovered, and the aim of this article is to investigate and compare the cohomology of the unipotent radical \(\mathfrak u\) of parabolic subalgebras over \(\mathbb C\) and \(\mathbb F_p\). Also a new proof of Kostant’s theorem and Polo-Tilouine’s extensions of the theorem is presented. The authors proof uses known linkage results in category \(\mathcal O_J\) and the graded \(G_1 T\) category for the first Frobenius kernel \(G_1\). Here \(J\) is a subset of the simple roots \(\Delta\) of the underlying root system.
When \(h\) is the Coxeter number for the underlying root system, the authors prove that when \(p<h-1\), there exists additional cohomology classes in \(H^\bullet(\mathfrak u,\mathbb F_p)\) beyond those given by Kostant’s formula. The proof relies on the modular representation theory of reductive algebraic groups. Also the authors exhibits natural classes that arise in \(H^{2p-1}(\mathfrak u,\bar{\mathbb F}_p)\) when the root system is \(\Phi=A_{p+1}\) which do not arise over fields of characteristic \(0\). In the end of the article, several low rank examples is given.
The authors give a relatively elementary definition of the category \(\mathcal O_J\), also introducing the parabolic Verma module, regular and singular weights, and they give a proposition allowing them to compare composition factors of the cohomology with coefficients in a module to the cohomology with trivial coefficients.
In a “parabolic computation”, the authors prove elementary results used to prove Kostant’s theorem and its generalization to prime characteristic. The prime characteristic case is much harder than the characteristic \(0\) case, because the control over the composition factors are much weaker.
Let \(L_j\) be the Levy factor of the parabolic subgroup relative to \(-J\). Then the first main result of the article is the proof of the following result:
Let \(J\subseteq\Delta\). Assume \(k=\mathbb C\) or \(k=\bar{\mathbb F}_p\) with \(p\geq h-1\). Then as an \(L_j\)-module \[ H^n(\mathfrak u_J,k)\cong{\underset{l(w)=n}{w\in{}^J W}}\oplus L_J(w\cdot 0) \] This result is used to prove the second main result which is the cohomology of \(\mathfrak u_J\) with coefficients in a finite-dimensional \(\mathfrak g\)-module.
The final main result is the converse of Kostant’s result, namely that there are extra cohomology classes that arise in characteristic \(p\) and \(p<h-1\).
In the last section of the article, tables over low rank examples is computed by the computer system MAGMA.
For the entire collection see [Zbl 1154.22002].