The second cohomology groups of nilpotent orbits in classical Lie algebras

Abstract

The second de Rham cohomology groups of nilpotent orbits in non-compact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in non-compact classical simple Lie algebras are computed; they are proven to be zero for nilpotent orbits in all the complex simple Lie algebras.A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes Theorem 3.3 of [BC1], may be of independent interest.2010 Mathematics Subject Classification. 57T15, 17B08. and a Haar measure µ J on J, define the integral J W (g)dµ J (g) ∈ Ω p (l) as follows:The above integral J W (g)dµ J (g) is also denoted by J W dµ J . The following equations are straightforward.For any a ∈ L,For any ω ∈ Ω p (l), from the left-invariance of the Haar measure µ J on J it follows thatLemma 3.1. Let L be a compact Lie group with Lie algebra l. Let p ≥ 1 be an integer.(1) If ω ∈ Ω p (l) is invariant then dω = 0.(2) Every element of H p (l, R) contains an unique invariant ω ∈ Ω p (l).(3) If J ⊂ L is a closed subgroup, then(4) If L is connected and ω ∈ Ω 2 (l), then ω ∈ Ω 2 (l) L if and only if ω ∈ Ω 2 (l/[l, l]). (1) is proved in [CE, p. 102, 12.3]. Statement (2) is proved in [CE, p. 102, Theorem 12.1]. Proof. StatementTo prove (3), note that it suffices to show thatLet µ J denote the Haar measure on J such that µ J (J) = 1. For any ω ∈ Ω p (l) J ∩ d(Ω p−1 (l)), we have ω = dν for some ν ∈ Ω p−1 (l). Now as ω is J-invariant, it follows that ω = Ad(g) * dν = dAd(g) * ν (3.4) for all g ∈ J. In particular, from (3.4) we have ω = J (dAd(g) * ν)dµ J (g) = d J (Ad(g) * ν)dµ J (g) .As µ J is preserved by the left multiplication by elements of J, it now follows that J (Ad(g) * ν)dµ J (g) ∈ Ω p−1 (l) J .This in turn implies that ω ∈ d(Ω p−1 (l) J ).

Homotopy type of the nilpotent orbits in classical Lie algebras

“…Subsequently, a few specialized notation are mentioned as and when they are needed. The notation and the background introduced in this section overlap with those in [BCM,§ 2] to some extent. However, for the sake of completeness and clarity of the exposition we also include them here.…”

“…Setting g := Lie G, let X ∈ g be a non-zero nilpotent element, and let O X be its adjoint G-orbit. A Lie theoretic reformulation of the second and the first cohomology groups of O X was obtained in [BCM,Theorem 4.2], incorporating a sl 2 (R)-triple containing X; the computations in [ChMa] also use this result crucially. Let {X, H, Y } be a sl 2 (R)triple in g, containing X, and let Z G (X, H, Y ) be the centralizer of the triple {X, H, Y } in G. Let K be a maximal compact subgroup in Z G (X, H, Y ), and M be a maximal compact subgroup in G containing K. Let m and z(k) be the Lie algebras of M and the center of K, respectively.…”

“…Let {X, H, Y } be a sl 2 (R)triple in g, containing X, and let Z G (X, H, Y ) be the centralizer of the triple {X, H, Y } in G. Let K be a maximal compact subgroup in Z G (X, H, Y ), and M be a maximal compact subgroup in G containing K. Let m and z(k) be the Lie algebras of M and the center of K, respectively. Then [BCM,Theorem 4.2] says that the computation of the second cohomology of the nilpotent orbits boils down to understanding the action of the component group K/K • on the subalgebra z(k) ∩ [m, m]. Thus, when M is semisimple, this amounts to describing the action of K/K • on z(k), and hence knowing the isomorphism class of K is enough to compute the second cohomology group in this case.…”

“…However, when M is not semisimple, it does not suffice to know the isomorphism classes of K and M, rather what is needed is an understand of the embedding of K in M. The case of su(p, q) dealt in [BCM,§ 4.4] constitutes a typical example of such a situation. Although the isomorphism class of M is a standard known object in the case of a non-compact classical simple real Lie group, and the isomorphism class of K can be obtained immediately using either the work of Springer and Steinberg [SpSt] (see also [BCM,Lemma 4.4]), hardly anything can be concluded, from these isomorphism classes, on how K is embedded in M. Consequently, one of the major objectives in [BCM] was to compute this embedding explicitly for all the nilpotent orbits in the classical real simple Lie algebras g for which maximal compact subgroup of G is not semisimple. This situation occurs in the cases when g is either su(p, q) or so * (2n) or sp(n, R).…”

“…In particular, the compact submanifold M/K of G/Z G (X) is in the same homotopy class as that of G/Z G (X); see Theorem 2.3 for details. Thus computations in [BCM] in fact yield compact sub-homogeneous spaces as convenient and optimal homotopy types of nilpotent orbits in the case when maximal compact subgroups of G are not semisimple (we refer to [BCM,Propositions 4.14,4.30 and 4.36]). It should be mentioned here that, for a certain technical reason, the homotopy types of the nilpotent orbits in so(p, q) were also described in [BCM,Proposition 4.22] under the following assumption on the partition associated to the parametrization:…”