Iwasawa Decomposition and Positivity

Abstract

The most basic of all the decompositions is the Iwasawa decomposition, which we introduce in the first section. The section computes appropriate Haar measures and Jacobians for the Iwasawa decomposition. For more similar Haar measure computations, see Chapter V, §3. In §3 we consider the Cartan Lie decomposition in connection with polynomial invariants. The last three sections are devoted to some linear algebra in connection with the notion of positivity (partial ordering) which will play an important role later. The polar decomposition is introduced in connection with a theorem of Harish-Chandra concerning this partial ordering. The theorem compares the size of the A-component in the Iwasawa and polar decompositions. Jacobian computations and basic effects of polar decomposition on differential operators will be given in Chapter VI.