Clifford algebras associated with symmetric pairs

Let g be a complex semisimple Lie algebra and theta be an involutive automorphism of g . Let g=k⊕ p be the decomposition of g into 1 and -1 eigenspaces of theta. Let spin nu : k → End(S) be the composite of nu : k→sop and the spin representation spin : sop → End(S), where nu( Y) = adgY&vbm0;p for Y ∈ k . A preliminary result of this thesis is to classify all the pairs ( g , theta) such that the corresponding spin nu 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 nu representation is primary. Then this representation has highest weight rho n. The main results in this thesis are the following decompositions. For the Clifford algebra we show that Cp ≅End&parl0;Vrn &parr0;⊗J as algebras, where J is isomorphic to a matrix algebra or a sum of two matrix algebras. Then we have ∧p=C ⊕D as a B0&vbm0;∧p -orthogonal direct sum for some non-degenerate symmetric bilinear form B0 on ∧g . Furthermore, we have ∧p=A∧C and C is minimal among all A-generating subspace in ∧p . 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,dia gg .