Bruhat, Cartan and Iwasawa decompositions in GLn(R), O(p, q) and GLn(Qp) / by Malak Adel Dbouk.

Abstract

Throughout my thesis I will discuss three main decompositions: the Bruhat, Cartan and Iwasawa decompositions. In the first chapter I will prove these decompositions in the general linear group of invertible matrices G. The main idea is that I have the subgroup of upper triangular matrices B of G, the maximal compact subgroup K of orthogonal matrices, A the subgroup of diagonal matrices, and the Weyl subgroup W. The Bruhat decomposition states that G is equal to the disjoint union of BwB where w belongs to W. Actually we can interpret the Bruhat decomposition in terms of flags. Given any two flags then there exists a basis of Rn that generates one of the flags and whose permutation generates the other flag. I use row and column operations to prove the result. The Iwasawa Decomposition states that G= KB. This says that every flag can be generated by an orthonormal basis. The Cartan decomposition states that G=KAK. I use the spectral theorem in addidtion to the fact that any symmetric positive definite matrix has a unique square root to prove this decomposition. In chapter 2, I prove these decompositions in the orthogonal group O(p,q). In this chapter the subgroup B is no more the subgroup of upper triangular matrices and A is no more the subgroup of diagonal matrices. In the proofs of the decompositions I use an isotropic basis of Rn under which the B and the A are easier to deal with. Another obstacle I face is making sure that the row and column operations I perform belong to O(p,q). This changes some aspects of the proof and makes it more tedious. Finally in chapter 3, I prove the decompositions in the general linear group os invertible matrices whose entries are in the field of p-adic numbers. In this chapter the maximal compact subgroup is the general linear group of invertible matrices whose entries belong to the ring of p-adic integers. A and B are still the subgroup of diagonal matrices and upper triangular matrices respectively. To prove the results we use row and column operations in addition to t