Resume of Representation Theory for Real Reductive Groups ( 19 May 2006 )

A reductive Lie algebra g is one that can be written C(g) ⊕ [g,g], where C(g) denotes the center of g. Equivalently, for any ideal a, there is another ideal b such that g = a⊕ b. A Cartan subalgebra of g is a subalgebra h that is maximal with respect to being abelian and having ad X being semisimple for all X ∈ h. For a reductive group, h = C(g) ⊕ h′, where h′ is a Cartan subalgebra of the semisimple group [g,g]. Fix a Cartan subalgebra h of g. For α ∈ h∗, define gα = {X ∈ g : [H,X] = α(H)X ∀H ∈ h}.