On related varieties to the commuting variety of a semisimple Lie algebra (Q360153)

Let \({\mathfrak g}\) be a finite-dimensional semisimple Lie algebra over an algebraic closed field of characteristic zero with \(G\) its adjoint group. Denote by \({{\mathfrak b}}_{{\mathfrak g}}\) and \(\text{rk}{\mathfrak g}\) be the dimensions of a Borel subalgebra and a Cartan subalgebra \({\mathfrak h}\) of \(\mathfrak g\), respectively. Let \(W\) be the Weyl group of \({\mathfrak g}\). Let \({\mathcal B}_{{\mathfrak g}}\) be the set of \((x,y)\) in \({\mathfrak g}\times{\mathfrak g}\) such that \(x\) and \(y\) are contained in the same Borel subalgebra, and let the nullcone, \({\mathcal N}\), be variety of pairs \((x,y)\) in \({\mathcal B}_{{\mathfrak g}}\) such that \(x\) and \(y\) are nilpotent.NEWLINENEWLINEThe author proved (Theorem 1.1) thatNEWLINENEWLINE(i) The variety \({\mathcal B}_{{\mathfrak g}}\) is closed in \({\mathfrak g}\times{\mathfrak g}\) and irreducible of dimension \(3{\mathfrak b}_{{\mathfrak g}}-\text{rk}{\mathfrak g}\), but it is not normal;NEWLINENEWLINE (ii) The algebra of \(W\)-invariant regular functions on \(\mathfrak h\times\mathfrak h\) is isomorphic to the algebra of \(G\)-invariant regular functions on \({\mathcal B}_{{\mathfrak g}}\).NEWLINENEWLINESimilarly, he proved (Theorem 1.2) that (i) The nullcone \({\mathcal N}\) is closed in \({\mathfrak g}\times{\mathfrak g}\) and irreducible of dimension \(3({\mathfrak b}_{{\mathfrak g}}-\text{rk}{\mathfrak g})\); (ii) The codimension of the set of its singular points is at least two and the normalization morphism of \({\mathcal N}\) is bijective.