Towards a symplectic version of the Chevalley restriction theorem

Abstract

If (G, V ) is a polar representation with Cartan subspace c and Weyl group W , it is shown that there is a natural morphism of Poisson schemes c ⊕ c * /W → V ⊕ V * / / /G. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if (G, V ) is visible. The conjecture is proved for visible stable locally free polar representations and some other examples.

On the normality of the null-fiber of the moment map for θ - and tori representations

“…Remark 4.13. Conjecture 1.3 in the case of a locally free stable θ-representation follows from [BLLT,Theorem 1.2]. We briefly explain here how to recover it using the results of this paper.…”

“…Thanks to the previous corollaries, we can re-interpret Proposition 3.5 as follows. The visible tori actions are essentially made of two part: a part of rank 0 provided by I f and several stable blocks of rank 1, each looking very much like [BLLT,Example 8.6]. Another appearence of stable blocks of rank 1 can be found in Lemma 4.9.…”

“…When k = C and (G, V ) is polar, we write c and c ∨ for dual Cartan subspaces of V and V * , following [BLLT,(3.4), (3.6)], together with dual decompositions…”

“…First, note that the 5 exceptional cases of Table 1 are of rank one. Then it follows from [BLLT,Proposition 3.4 & 5.1] that the natural map c ⊕ c ∨ /W → V ⊕ V * / / /G is a dominant closed immersion, hence an isomorphism. In the other cases, Theorem 4.6 ensures that µ −1 (0) is normal.…”

“…Since it is made of semisimple elements, the representation is stable. (iv) follows from [BLLT,Lemma 6.3].…”