An exotic Deligne-Langlands correspondence for symplectic groups

Abstract

Let $G=\mathrm{Sp}(2n,\mathbb{C})$ be a complex symplectic group. We introduce a ($G\times ({\mathbb{C}}^{\times }{)}^{\ell +1}$)-variety ${\mathfrak{N}}_{\ell }$, which we call the $\ell$-exotic nilpotent cone. Then, we realize the Hecke algebra $\mathbb{H}$ of type $C_n^{(1)}$ with three parameters via equivariant algebraic $K$-theory in terms of the geometry of ${\mathfrak{N}}_2$. This enables us to establish a Deligne-Langlands–type classification of simple $\mathbb{H}$-modules under a mild assumption on parameters. As applications, we present a character formula and multiplicity formulas of $\mathbb{H}$-modules