A comparison of Weyl group actions on Lagrangian cycles
Let \(G\) be a connected complex reductive algebraic group. Write \(W\) for its Weyl group and \(X\) for its flag variety. In this paper, \(Q\) is any real closed Lie subgroup of \(G\) having finitely many orbits on \(X\). For such \(Q\), write \(T^*_Q(X)\) for the conormal variety of the \(Q\)-action. In this case, the cycles supported by \(T^*_Q(X)\) may be given a \(W\)-module structure (e.g., see Lusztig). Moreover, write \(D_Q^b(X)\) for the bounded derived category of sheaves on \(X\) whose characteristic variety is contained in \(T^*_Q(X)\). Then it is also true that the Grothendieck group \({\mathcal K}_Q(X)\) of \(D_Q^b(X)\) has a \(W\)-module structure given by intertwining operators. The main theorem of this paper proves the characteristic cycle and defines a homomorphism of \(W\)-modules mapping \({\mathcal K}_Q(X) \rightarrow H_{2n}(T^*_Q(X), {\mathbb C})\), where \(n=\dim_{\mathbb C}X\) and \(H_*(\cdot, {\mathbb C})\) is the Borel-Moore homology.
connected complex reductive algebraic group
scientific article; zbMATH DE number 1251259
scientific article