By Luther Pfahler Eisenhart

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E of N I onto N I . e c G I pI is a topological embedding. Under the map gP -+ gpI, each flag manifold is the image of G I P. This flag manifold, often denoted by F, is of central importance in this book and is called the Furstenberg boundary. It has a cellular decomposition, corresponding to the cellular decomposition of G (see [Gl, pp. 76-81]), that can often be can be used to reduce calculations in flag manifolds to calculations of affine type. 20. Proposition. (The cellular Bruhat decomposition) If w E W.

0 and ka2a· 0 = a2ka . o. Hence, ka . 0 = a . 0 and so k E aKa -1. However, since a· 0 E Xl and k E KI M, it follows that if k = fm, f E K I, m EM, then fa . 0 = a . 0 and so f E aK I a-I. 0 As in the case of a polyhedral cone decomposition, the fundamental sequences determine a unique compactification of X (equivalently, of p). 38. Theorem. There is a unique compactification X of X such that (1) every fundamental sequence converges in X; and (2) the limits of two fundamental sequences in X agree if and only if their formal limits agree.

28 III. GEOMETRICAL CONSTRUCTIONS OF COMPACTIFICATIONS Proof. In [B4, p. 245J it is shown that, for any geodesic 'Y from 0 directed by a unit vector H in a+, [an· 'YJ = bJ if an E AN. Consequently, for such a geodesic, if [g. 16), it is clear that pI stabilizes 'Y if 'Y(t) = exptH '0, HE GI. The stabilizer of bJ is therefore a standard parabolic subgroup pI that contains pl. If [k· 'YJ = b]' then d(exptAd(k)H· o,exptH· o),t > 0 is bounded. In view of the non-positivity of the curvature of X this is only possible if Ad(k)H = H.