Springer Correspondence

Setup

Let \(G\) be a connected reductive group over an algebraically closed field \(k\), with Weyl group \(W = N_G(T)/T\). Let \(\mathcal{B} = G/B\) be the flag variety and \(\mathcal{N} \subset \mathrm{Lie}(G)\) the nilpotent cone.

The Springer resolution

The Springer resolution is the map \[ \mu : \widetilde{\mathcal{N}} \longrightarrow \mathcal{N}, \] where \[ \widetilde{\mathcal{N}} = \{(X, B') : X \in \mathrm{Lie}(B'),\; B' \in \mathcal{B}\} \subset \mathcal{N} \times \mathcal{B}, \] and \(\mu(X, B') = X\). This is a proper, \(G\)-equivariant, birational morphism resolving the singularities of \(\mathcal{N}\).

The W-action and the correspondence

Springer showed that the Weyl group \(W\) acts on each cohomology group \(H^i(\mu^{-1}(X), \overline{\mathbb{Q}}_\ell)\) for \(X \in \mathcal{N}\). Specifically:

Springer's theorem. For each nilpotent orbit \(\mathcal{O} \subset \mathcal{N}\) and each \(G\)-equivariant local system \(\mathcal{L}\) on \(\mathcal{O}\), there exists an irreducible representation \(\phi_{\mathcal{O},\mathcal{L}}\) of \(W\) appearing in \(H^{2d}(\mu^{-1}(X), \overline{\mathbb{Q}}_\ell)\) for \(X \in \mathcal{O}\) and \(d = \dim \mu^{-1}(X)\).

This gives the Springer correspondence: \[ \bigsqcup_{\mathcal{O}} \{\text{irred.\ }G\text{-equivariant local systems on }\mathcal{O}\} \xrightarrow{\;\sim\;} \widehat{W}, \] a bijection from pairs \((\mathcal{O}, \mathcal{L})\) to the set \(\widehat{W}\) of irreducible complex representations of \(W\).

For the trivial local system

For the trivial local system on each orbit, the Springer correspondence gives a map (not a bijection): \[ \{\text{nilpotent orbits}\} \to \widehat{W}, \quad \mathcal{O} \mapsto \phi_{\mathcal{O}}. \] In type \(A_{n-1}\) (\(G = \mathrm{SL}_n\)), this recovers the Robinson-Schensted correspondence: both sides are parametrised by partitions of \(n\).

Geometric interpretation

Via the decomposition theorem for \(\mu_*(\overline{\mathbb{Q}}_\ell)\), the Springer correspondence can be expressed as: \[ \mu_*(\overline{\mathbb{Q}}_\ell)[\dim G] \cong \bigoplus_{(\mathcal{O},\mathcal{L})} V_{\mathcal{O},\mathcal{L}} \otimes \mathrm{IC}(\overline{\mathcal{O}}, \mathcal{L}), \] where \(V_{\mathcal{O},\mathcal{L}}\) is the corresponding \(W\)-representation and \(\mathrm{IC}\) denotes the intersection complex.