The Images of Restriction Maps and Dirac Cohomology

Abstract

Let G be a real semisimple Lie group with finite center. Let 𝔤 0 = 𝔨 0 ⊕ 𝔭 0 be a Cartan decomposition for the Lie algebra 𝔤 0 of G. Let 𝔤 (resp., 𝔨, 𝔭) be the complexification of 𝔤 0 (resp., 𝔨 0, 𝔭 0). Let 𝔥 0 = 𝔱 0 ⊕ 𝔞 0 be a fundamental Cartan subalgebra of 𝔤 0. Then 𝔱 0 ⊂ 𝔥 0 is a Cartan subalgebra of 𝔨 0. Let Z(𝔨) and Z(𝔤) be, respectively, the centers of the universal enveloping algebras of 𝔨 and 𝔤. There is a homomorphism ζ: Z(𝔤) → Z(𝔨), which plays an important role in Huang and Pandžić's proof of a conjecture of Vogan on Dirac cohomology. The map ζ is defined by the restriction map via Harish–Chandra isomorphisms, where W(𝔤,𝔥) and W(𝔨,𝔱) are the Weyl groups of 𝔤 and 𝔨, respectively. Kostant generalizes the result of Huang and Pandžić to the case where 𝔯 is an arbitrary reductive subalgebra of 𝔤. In Kostant's work, the map where 𝔱 is a Cartan subalgebra of 𝔯 contained in 𝔥, plays a similar role as the restriction map described above. In this article we determine the images of restriction maps for symmetric pairs and reductive pairs with closer Coxeter numbers. Determining the images of these restriction maps is useful for calculating Dirac cohomology of 𝔤-modules, cohomology of homogeneous spaces and invariant differential operators on G-manifolds.