Let G be a real semisimple Lie group with finite center. Let g₀ = k₀ ⊕ p₀ be a Cartan decomposition for the Lie algebra g₀ of G. Let g (resp., k, p) be the complexification of g₀ (resp., k₀, P₀). Let h₀ = t₀ ⊕ a₀ be a fundamental Cartan subalgebra of g₀. Then t₀ ⊂ h₀ is a Cartan subalgebra of k₀. 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 _h/t ^{W(g,h) W(k,t)}...[ Read more ]

Let G be a real semisimple Lie group with finite center. Let g₀ = k₀ ⊕ p₀ be a Cartan decomposition for the Lie algebra g₀ of G. Let g (resp., k, p) be the complexification of g₀ (resp., k₀, P₀). Let h₀ = t₀ ⊕ a₀ be a fundamental Cartan subalgebra of g₀. Then t₀ ⊂ h₀ is a Cartan subalgebra of k₀. 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

Res_h/t : S(h)^W(g,h) → S(t)^W(k,t)

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

Res_h/t : S(h)^W(g,h) → S(t)^W(r,t),

where t is a Cartan subalgebra of r contained in h, plays a similar role as the restriction map described above. To determine the image of the restriction map for arbitrary reductive pairs seems to be a formidable task. However, we manage to solve this problem for reductive pairs with closer Coxeter numbers.

In this thesis we determine the images of restriction maps for symmetric pairs and reductive pairs with closer Coxter 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.