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 ).

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Abstract. In [BC], the second de Rham cohomology groups of nilpotent orbits in all the complex simple Lie algebras are described. In this paper we consider non-compact non-complex exceptional Lie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotent orbits. For the rest of cases of nilpotent orbits, which are not covered in the above computations, we obtain upper bounds for the dimensions of the second cohomology groups.

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