Gauss decomposition

of a topological group $ G $

A representation of an everywhere-dense subset $ G _ {0} \subset G $ in the form $ G _ {0} = NH N ^ {*} $, where $ H $ is an Abelian subgroup of $ G $, and $ N $ and $ N ^ {*} $ are nilpotent groups of $ G $, normalized by $ H $. If $ G $ is the group $ \mathop{\rm GL} ( m, \mathbf R ) $ of non-singular real matrices of order $ m $, $ H $ is the subgroup of diagonal matrices, $ N $( respectively, $ N ^ {*} $) is the subgroup of lower-triangular (upper-triangular) matrices with unit elements on the principal diagonal, and $ G _ {0} $ is the subset of matrices in $ G $ whose principal minors are non-zero, then the decomposition $ G _ {0} = NH N ^ {*} $ is known as the Gauss decomposition of the general linear group and is directly connected with the Gauss method for the solution of systems of linear equations: If $ g _ {0} = nh n ^ {*} $, where $ n \in N $, $ h \in H $, $ n ^ {*} \in N ^ {*} $, is the non-singular coefficient matrix of the system of linear equations $ g _ {0} x = b $, then it may be converted by the Gauss method into the triangular form $ h n ^ {*} $ by multiplying it from the left by the lower-triangular matrix $ n ^ {-} 1 $, $ n \in N $. A rigorous definition of the Gauss decomposition necessitates the introduction of the following terms. Let $ G $ be a topological group, let $ H $ be a subgroup of it, and let $ N $ and $ N ^ {*} $ be nilpotent subgroups in $ G $, normalized by $ H $. The subgroup $ H $ is called a triangular truncation of $ G $ if: 1) $ N \in D( R) $, $ N ^ {*} \subset D( R ^ {*} ) $, where $ D( X) $ is the commutator subgroup of the group $ X $ and $ R $ and $ R ^ {*} $ are connected solvable subgroups of $ G $; and 2) the set $ G _ {0} = NH N ^ {*} $ is everywhere dense in $ G $, and the decomposition $ NH N ^ {*} $ is unique. The decomposition $ G _ {0} = NH N ^ {*} $ is called a triangular decomposition in $ G $. If $ H $ is an Abelian group, this decomposition is called a completely-triangular decomposition or a Gauss decomposition. In such a case the subgroups $ B = NH = HN $, $ B ^ {*} = N ^ {*} H = H N ^ {*} $ are solvable. Let $ \pi $ be an irreducible (continuous) representation of $ G $ in a finite-dimensional vector space $ V $, and let $ V _ {0} $ be the subspace of all vectors in $ V $ which are fixed with respect to $ N ^ {*} $; $ V _ {0} $ will then be invariant with respect to $ H $, while the representation $ \alpha $ of $ H $ on $ V _ {0} $ will be irreducible. The representation $ \alpha $ unambiguously defines $ \pi $, up to an equivalence. Let $ \alpha $ also denote the representation of $ B $ on $ V _ {0} $ restricting to $ \alpha $ on $ H $ and being trivial on $ N $. Let $ e ( \alpha ) $ denote the representation of $ G $ on the vector space $ C ( G , V _ {0} ) $ induced by this $ \alpha $. Then $ \pi $ is contained (as an invariant part) in $ e ( \alpha ) $, and the space $ \mathop{\rm Hom} _ {G} ( \pi , e( \alpha )) $ is one-dimensional. If $ H $ is an Abelian subgroup, then $ V _ {0} $ is one-dimensional and $ \alpha $ is a character of the group $ H $. The following examples of triangular decompositions of Lie groups are known. 1) Let $ G $ be a reductive connected complex Lie group with Cartan subalgebra $ H _ {0} $ and let $ H $ be a reductive connected subgroup in $ G $ containing $ H _ {0} $. The subgroup $ H $ is then a triangular truncation of $ G $. 2) Let $ G $ be a reductive connected linear Lie group; $ G $ will then contain a triangular truncation $ H = MA $, where $ A $ is a simply-connected Abelian subgroup in $ G $( generated by the non-compact roots in the Lie algebra of $ G $), and $ M $ is the centralizer of $ A $ in the maximal compact subgroup $ K \subset G $. 3) In particular, any reductive connected complex Lie group permits a Gauss decomposition $ G _ {0} = NH N ^ {*} $, where $ H $ is the Cartan subgroup of $ G $ and $ N $( respectively $ N ^ {*} $) is an analytic subgroup in $ G $ whose Lie algebra is spanned by all root vectors $ e _ \alpha $, $ \alpha < 0 $( respectively $ \alpha > 0 $), with $ \alpha $ denoting the roots with respect to $ H $, i.e. $ HN $ and $ H N ^ {*} $ are opposite Borel subgroups (cf. Borel subgroup). In examples 1)–3) the subgroups $ N $ and $ N ^ {*} $ are simply connected, $ G _ {0} $ is open in $ G $ in the Zariski topology, while the mapping $ N \times H \times N ^ {*} $, $ ( n, h, n ^ {*} ) \rightarrow nh n ^ {*} $, is an isomorphism of algebraic varieties (and, in particular, a homeomorphism). This implies that the algebraic variety $ G $ is rational.

References

[1]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)