Abelian ideals and cohomology of symplectic type

Abstract:

Let $\mathfrak {b}$ be a Borel subalgebra of the symplectic Lie algebra $\mathfrak {sp}(2n,\mathbb {C})$ and let $\mathfrak {n}$ be the corresponding maximal nilpotent subalgebra. We find a connection between the abelian ideals of $\mathfrak {b}$ and the cohomology of $\mathfrak {n}$ with trivial coefficients. Using this connection, we are able to enumerate the number of abelian ideals of $\mathfrak {b}$ with given dimension via the Poincaré polynomials of Weyl groups of types $A_{n-1}$ and $C_n$.