The homology of the Steinberg variety and Weyl group coinvariants

Let G be a complex, connected, reductive algebraic group with Weyl group W and Steinberg variety Z. We show that the graded Borel-Moore homology of Z is isomorphic to the smash product of the coinvariant algebra of W and the group algebra of W.

💡 Research Summary

The paper investigates the Borel‑Moore homology of the Steinberg variety Z associated with a complex connected reductive algebraic group G, whose Weyl group is W. After recalling the definition of Z as the fibre product 𝔅×_{𝔤}𝔅 (where 𝔅=G/B is the flag variety and 𝔤 the Lie algebra), the authors establish a cell decomposition indexed by the elements of W. Each cell Z_w has complex dimension equal to the length ℓ(w) and contributes a fundamental class