Schubert cells and representation theory
Let \(G\) be a connected real reductive group and \(P\) its parabolic subgroup. The goal of the paper is to give procedures for calculating the integer homology and cohomology groups of the real flag manifold \(G/P\) in terms of its Schubert cells. Let \(W_G\) and \(W_P\) denote the Weyl groups of the complexifications of \(G\) and of the Levi subgroup of \(P\) and let \(\widetilde W^P\) be the set of the minimal length representatives of left cosets of \(W_G\) modulo \(W_P\). The Bruhat decomposition of \(G\) gives rise to the decomposition of \(G/P\) into the Schubert cells \(C_w\) parametrized by \(w\in\widetilde W^P\), such that \(\dim C_w = l(w)\). Chain and cochain complexes for calculating \(H_*(G/P,\mathbf Z)\) and \(H^*(G/P,\mathbf Z)\) are constructed. E.g., to calculate \(H^*(G/P,\mathbf Z)\), one may use the complex \(S^*\) with \[ S^k = \bigoplus_{\substack{ w\in\widetilde W^P\\ l(w)=k}}\mathbf Z_w, \] whose coboundary operators are given by matrices with entries in \(\{0,\pm 2\}\) expressed in terms of the Weyl groups. Topologically, these entries contain information concerning non-orientability of certain submanifolds of \(G/P\). The simplest case is when \(G\) possesses the following property: for every real root \(\alpha\) of \(G\) with respect to its maximally split Cartan subgroup \(H\), the element \(\Phi_{\alpha}(-I)\) is contained in the identity component of \(H\), where \(\Phi_{\alpha}: SL(2,\mathbf R)\to G\) is the homomorphism associated with \(\alpha\). Then, for all parabolic subgroups \(P\subset G\), all the coboundaries are 0, and hence \(H^k(G/P,\mathbf Z)\simeq\mathbf Z^{s_k}\), where \(s_k\) is the number of elements of \(\widetilde W^P\) of length \(k\). Another result is a necessary and sufficient condition for a Schubert cell \(C_w\) to be a cycle; in this case, it always determines a non-zero element of \(H_{l(w)}(G/P,\mathbf Z)\). The proofs exploit perverse sheaves and principal series representations.
scientific article; zbMATH DE number 1336938