On Kostant’s Partial Order on Hyperbolic Elements

We study Kostant’s partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse statement of [3, Theorem 6.1] on hyperbolic elements. A matrix in GLn(C) is called elliptic (resp. hyperbolic) if it is diagonalizable with norm 1 (resp. real positive) eigenvalues. It is called unipotent if all its eigenvalues are 1. The complete multiplicative Jordan decomposition of g ∈ GLn(C) asserts that g = ehu for e, h, u ∈ GLn(C), where e is elliptic, h is hyperbolic, u is unipotent, and these three elements commute (cf. [2, p430-431]). The decomposition can be easily seen when g is in a Jordan canonical form: if the diagonal entries (i.e. eigenvalues) of the Jordan canonical form are z1, · · · , zn, then (1) e = diag ( z1 |z1| , · · · , zn |zn| ) , h = diag (|z1|, · · · , |zn|) , and u = heg is an upper triangular matrix with diagonal entries 1. The above decomposition can be extended to semisimple Lie groups. Let G be a connected real semisimple Lie group with Lie algebra g. An element e ∈ G is elliptic if Ad e ∈ Aut g is diagonalizable over C with eigenvalues of modulus 1. An element h ∈ G is called hyperbolic if h = expX where X ∈ g is real semisimple, that is, adX ∈ End g is diagonalizable over R with real eigenvalues. An element u ∈ G is called unipotent if u = expX where X ∈ g is nilpotent, that is, adX ∈ End g is nilpotent. The complete multiplicative Jordan decomposition [3, Proposition 2.1] for G asserts that each g ∈ G can be uniquely written as (2) g = ehu, 2000 Mathematics Subject Classification: Primary 22E46