THESIS
2004
Abstract
Let g be a complex semisimple Lie algebra and θ be an involutive automorphism of g. Let g = k ⊕ p be the decomposition of g into 1 and -1 eigenspaces of θ. Let spin v : k → End(S) be the composite of v : k→so(p) and the spin representation spin : so(p) → End(S), where v(Y) = ad
_{g}Y subp
_{n}...[
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Let g be a complex semisimple Lie algebra and θ be an involutive automorphism of g. Let g = k ⊕ p be the decomposition of g into 1 and -1 eigenspaces of θ. Let spin v : k → End(S) be the composite of v : k→so(p) and the spin representation spin : so(p) → End(S), where v(Y) = ad
_{g}Y sub>p for Y ∈ k. A preliminary result of this thesis is to classify all the pairs (g , θ) such that the corresponding spin v representation is primary. This result is also obtained by D. Panyushev using a different method, but our argument is considerably short.
Suppose that (g, k) is a symmetric pair such that the corresponding spin v representation is primary. Then this representation has highest weight ρ
_{n}. The main results in this thesis are the following decompositions. For the Clifford algebra we show that C(p) ≅ End(V
_{ρn}) ⊗ J as algebras, where J is isomorphic to a matrix algebra or a sum of two matrix algebras. Then we have Λp = C ⊕ D as a B
_{0} _{Λp}-orthogonal direct sum for some non-degenerate symmetric bilinear form B
_{0} on Λg. Furthermore, we have Λp = AΛC and C is minimal among all A-genrating subspace in Λp. These decompositions are generalization of B. Konstant's results on a semisimple Lie algebra g, which can be regarded as a special case of symmetric pair (g ⊕ g, diag(g)).
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