Orbits in the Enhanced and Exotic Nilpotent Cones

We give a semi-direct product decomposition of the point stabilisers for the enhanced and exotic nilpotent cones. In particular, we arrive at formulas for the number of points in each orbit over a finite field. This is in accordance with a conjecture of Achar-Henderson. Introduction In the theory of algebraic groups, we find that there is much insight to be gained from studying the nilpotent cone N of an algebraic group G, which consists of the nilpotent elements in the Lie algebra g of G. One of the main reasons for this is because G acts on N , usually by conjugation, and there is an injective map (when G is reductive) from the G-orbits in N to the irreducible representations of the Weyl group W of G. This is a consequence of the Springer correspondence and was originally discovered by Springer [7] in 1976 and was explicitly described in all cases by Lusztig and Shoji by the early 1980s (see for example Shoji [6]). A well-known example of the Springer correspondence can be seen for the group G = GL(V ) of invertible endomorphisms of an n-dimensional vector space V over an algebraically closed field k. This is a reductive group whose Lie algebra is gln = End(V ) (the endomorphisms of V ) and N is the nilpotent endomorphisms. Since G acts by conjugation on N , we have that the G-orbits in N are in bijection with Pn, the partitions of n, by the Jordan canonical form theorem. On the other hand, the Weyl group of G is just the Symmetric group Sn, whose irreducible representations are known to be in bijection with Pn also. So in this case the G-orbits in N are actually in bijection with the irreducible representations of W . However the same cannot be said for groups such as K = Sp(W ), where W is a 2n-dimensional symplectic space over k, because the map fails to be surjective. 1