An exotic nilpotent cone for symplectic group

Let G be a complex symplectic group. We construct a certain Gvariety N, which shares many nice properties with the nilpotent cone of G. For example, it is normal, has finitely many G-orbits, and admits a G-equivariant resolution of singularity in a similar way to the Springer resolution. Our new G-variety admits an action of G× C × C instead of the G×C-action on the nilpotent cone. This enables us to realize the Hecke algebra with unequal parameters via equivariant algebraic K-theory in the companion paper [math.RT/0601155]. Introduction and main results Let G = Sp(2n,C) be a complex symplectic group of rank n. Let B be its Borel subgroup. Then, we have the moment map μS : T (G/B) −→ NG, where NG ⊂ g is the nilpotent cone of G. This map, sometimes called the Springer resolution, is very important in representation theory since it is deeply connected with the Springer representation of the Weyl group and the Lusztig realization of the Hecke algebra (cf. [CG97] and [KL87]). In this paper, we construct a certain variant N of NG which shares many nice properties with NG. It is even “better” in the sense that it admits two mutually commutative C-actions. Let T be a maximal torus of B. We denote by X(T ) the character group of T . Let R be the root system of (G, T ) and let R be its positive part defined by B. We embed R and R into a n-dimensional Euclid space E = ⊕iCǫi as: R = {ǫi ± ǫj}i<j ∪ {2ǫi} ⊂ {±ǫi ± ǫj} ∪ {±2ǫi} = R ⊂ E. We define V1 := C 2n and V2 := (∧V1)/C. These representations have B-highest weights ǫ1 and ǫ1 + ǫ2, respectively. We put V := V1 ⊕ V2 and call it the exotic representation of Sp(2n). For a G-module V , we define its weight λ-part (with respect to T ) as V [λ]. The positive part V + of V is defined as V + := ⊕