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| Title: | Restrictions of invariants of reflections and dirac cohomology |
| Authors: | Cheng, Jian-Jun |
| Issue Date: | 2004 |
| Abstract: | 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
Resh/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
Resh/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. |
| Description: | Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2004
viii, 50 leaves ; 30 cm
HKUST Call Number: Thesis MATH 2004 Cheng |
| URI: | http://hdl.handle.net/1783.1/2013 |
| Appears in Collections: | MATH Doctoral Theses
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