Chevalley restriction theorem for the cyclic quiver

We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. The aim of this paper is to prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. When the quiver is of type Â0, we recover the results for gln. The proof of our Chevalley restriction theorem is similar to the proof for gln; however, the proof of the double analogue uses a theorem of Crawley-Boevey on decomposition of quiver varieties. The double analogue is the limiting case of an isomorphism between a Calogero-Moser space and the center of a symplectic reflection algebra proved by Etingof and Ginzburg. It is also the associated graded version of a conjectural Harish-Chandra isomorphism for the cyclic quiver. We now introduce our notations. Let Q be the cyclic quiver with m vertices. Let δ = (1, . . . , 1) be the minimal positive imaginary root. Let Rn = Rep(Q,nδ) be the space of representations of Q with dimension vector nδ. Thus, Rn = gln × · · · × gln } {{ } m . Next, let h be the subspace of diagonal matrices in gln, and let Ln = {(z, . . . , z) ∈ Rn | z ∈ h}. Note that Ln is a n dimensional subspace of Rn. Let Gn = GLn × · · · ×GLn } {{ } m . An element (g1, . . . , gm) ∈ Gn acts on an element (x1, . . . , xm) ∈ Rn, giving (g 2 x1g1, g −1 3 x2g2, . . . , g −1 1 xmgm). Let Sn be the symmetric group on n letters, which we will also regard as the subgroup of permutation matrices in GLn. Finally, let Wn = Sn ⋉ (Z/mZ) .