(1) Cartan decomposition

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This is the simplest case Given a semisimple Lie group G, its real Lie algebra g always possesses a Cartan involution, that is, an automorphism : g g, with square equal to the identity, [X , Y ] = [ (X ), (Y )] , X,Y g, 2 = I d , (B16)

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and such that the symmetric bilinear form B (X , Y ) = B(X , Y ) is positive-definite, where B is the Cartan Killing form Then the Cartan involution yields an eigenspace decomposition

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g=k p

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(B17)

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of g into +1 and 1 eigenspaces It follows that [k, k] k , [k, p] p , [p, p] k (B18)

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Assume for simplicity that the center of G is nite, and let K denote the analytical subgroup of G with Lie algebra k Then: K is closed and maximal-compact, there exists a Lie group automorphism of G, with differential , such that 2 = I d and the subgroup xed by is K, the mapping K ~ p G given by (k, X ) k exp X is a diffeomorphism One may also write the diffeomorphism as (X , k) exp X k = p k

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B3 Lie Groups

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(2) Iwasawa decomposition

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Any connected semisimple Lie group G has an Iwasawa decomposition into three closed subgroups, namely, G = K AN , where K is a maximal compact subgroup, A is Abelian, and N nilpotent, and the last two are simply connected This means that every element g G admits a unique factorization g = kan, k K , a A, n N , and the multiplication map K ~ A ~ N G given by (k, a, n) kan is a diffeomorphism Assume that G has a nite center Let M be the centralizer of A in K, that is, M = {k K : ka = ak, a A} (if the center of G is not nite, the de nition of M is slightly more involved) Then P = MAN is a closed subgroup of G, called the minimal parabolic subgroup The interest in this subgroup is that the quotient manifold X = G/P ~ K /M carries the unitary irreducible representations of the principal series of G (which are induced representations), in the sense that these representations are realized in the Hilbert space L2 (X , ), with the natural G-invariant measure To give a concrete example, take G = SO o (3, 1), the Lorentz group Then the Iwasawa decomposition reads SO o (3, 1) = SO(3) A N , (B19)

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where A ~ SO o (1, 1) ~ R is the subgroup of Lorentz boosts in the z-direction and N ~ C is two-dimensional and Abelian Then M = SO(2), the subgroup of rotations around the z-axis, so that X = G/P ~ SO(3)/SO(2) ~ S 2 , the 2-sphere A closely related decomposition is the so-called K AK decomposition Let G again be a semisimple Lie group with a nite center, K a maximal compact subgroup, and G = K AN the corresponding Iwasawa decomposition Then every element in G has a decomposition as k 1 ak 2 with k 1 , k 2 K and a A This decomposition is in general not unique, but a is unique up to conjugation A familiar example of a K AK decomposition is the expression of a general rotation SO(3) as the product of three rotations, parameterized by the Euler angles: = m( ) u( ) m( ) , (B20)

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where m and u denote rotations around the z-axis and the y-axis, respectively

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(3) Gauss decomposition

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Let G again be a semisimple Lie group and G c = exp gc the corresponding complexi ed group If b is a subalgebra of gc , we call it maximal (in the sense of Perelomov [10]) if b b = gc , where b is the conjugate of b in gc Let {H j , E } be a Cartan Chevalley basis of the complexi ed Lie algebra gc Then G c possesses remarkable subgroups: H c , the Cartan subgroup generated by {H j } B , the Borel subgroups, which are maximal connected solvable subgroups, corresponding to the subalgebras b , generated by {H j , E | }; if b is maximal, then b+ = b and b = b generate Borel subgroups

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