Harmonic polynomials and Springer's representation for \(\text{SL}(n,\mathbb{C})\)
Let \(G\) be a connected reductive complex algebraic group and let \(\mathfrak g\) be its Lie algebra. Let \(B\) be the variety of all Borel subgroups of \(G\). This paper only deals with the case \({\mathfrak g} = \text{sl}(n,\mathbb{C})\). In the special case \({\mathfrak g} = \text{sl}(n,\mathbb{C})\), \(B\) becomes isomorphic to the variety \(T = T(V)\) of all complete flags in an \(n\)-dimensional complex space \(V\) and the nilpotent orbits are in bijective correspondence with partitions of \(n\). Let \(u\) be a unipotent linear transformation of \(V\) of Jordan partition \(\sigma = (\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_m)\). Then the Springer variety, denoted by \(T_\sigma\), is the fixed point subvariety of \(T\) with respect to the action of \(u\). Let \(\Delta\) be the root system of the Weyl group \(S_\mu = S_{\mu_1} \times \dots \times S_{\mu_r}\) and \(\mu = (\mu_1 \geq \dots \geq \mu_r)\). Clearly, for the Weyl groups \(S_\mu\), the Weyl dimension polynomial \(p_\Delta\) is \(S_\mu\)-harmonic. Using results of \textit{S. G. Hulsurkar} [Invent. Math. 27, 45-52 (1974; Zbl 0298.17005)], an explicit expression of the homology \(H_* (T_\sigma, \mathbb{Q})\) is given in terms of the Weyl dimension polynomials \(p_\Delta\).
connected reductive complex algebraic group
scientific article; zbMATH DE number 130532
scientific article