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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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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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and 13 million people died, with military costs and property destruction the equivalent of trillions of dollars The bleak intrawar years brought further protectionist measures As tempers ared among trading partners, a worldwide depression set in, allowing for the rise of fascist and nationalist movements, and ultimately leading to World War II The Second World War s death toll totaled between 50 and 70 million including fatalities from famine and disease3 More than $1 trillion was spent by countries in combat and an estimated $3 trillion in property was destroyed4 Not until well after World War II did trade begin to approximate the late nineteenth-century volume

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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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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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