Kostant's conjecture and the geometry of Cartan subalgebras in the rank two and exceptional cases (Q1922466)

Motivated by a conjecture of Kostant (now proved by Serre) on the existence of copies of \(PSL(2,q)\) in a simple complex Lie group \(G\) with Coxeter number \(h\) such that \(2h+1 =q\) is an odd prime power, the author studies decompositions of complex simple Lie algebras \({\mathfrak g}\) as vector space direct sums of Cartan subalgebras. More precisely, one can show that there is a Cartan subalgebra \({\mathfrak h}\) of \({\mathfrak g}\) such that a Borel subgroup of \(PSL(2,q)\) fixes \({\mathfrak h}\). Then one has \(q+1= 2(h+1)\) conjugates of \({\mathfrak h}\) under \(PSL(2,q)\). On the other hand one has \(\dim_\mathbb{C} {\mathfrak g}= (h+1) \dim_\mathbb{C} {\mathfrak h}\), so what one wants is to pick half of the Cartan algebras in the \(PSL(2,q)\)-orbit of \({\mathfrak h}\) such that \({\mathfrak g}\) is the direct sum of these. The author shows that this can in fact be done in the case of rank 2 and the exceptional Lie algebras. A key role in his construction is played by the elements in \(PSL(2,q)\) of order \(h\) (Kostant elements) or \(h+1\) (Kac elements). The main tools used come from classical number theory.