The torus-equivariant cohomology of nilpotent orbits (Q2791849)

The study of nilpotent orbits lies at the interface of algebraic geometry, representation theory, and symplectic geometry. Indeed, we have the famous Springer resolution \(\mu:T^\ast(G/B)\rightarrow\mathcal N\) of the singular nilpotent cone. The fibres of \(\mu\) over a given nilpotent orbit \(\mathcal O\) are isomorphic as complex varieties, and this isomorphism class is called the Springer fibre of \(\mathcal O\). The Springer correspondence then gives a realization of the irreducible complex Weyl group representations on the Borel-Moore homology groups of the Springer fibres. From the symplectic standpoint, we note that coadjoint \(G\)-orbits are canonically complex symplectic manifolds. Since the Killing form on the Lie algebra \(\mathfrak g\) provides an isomorphism between the adjoint and coadjoint representations of \(G\), it follows that adjoint \(G\)-orbits (and in particular, nilpotent \(G\)-orbits) are naturally complex symplectic manifolds. Some attention has also been given to the matter of computing topological invariants of nilpotent orbits. Indeed, the work of Springer, Steinberg, and others has led to a computation of the fundamental group of every nilpotent orbit in the classical Lie algebras. Also, Juteau's paper gives the integral cohomology groups of the minimal nilpotent orbit in each of the finite-dimensional complex simple Lie algebras. Additionally, Biswas and Chatterjee have computed \(H^2(\mathcal O;\mathbb R)\) for \(\mathcal O\) any nilpotent orbit in a finite-dimensional complex simple Lie algebra.NEWLINENEWLINEIn the paper under review, the authors are doing the computation of the \(T\)-equivariant cohomology algebras of the \(G\)-orbits \(\mathcal O_{\mathrm{reg}}\) and \(\mathcal O_{\min}\).