Irreducible components of the nilpotent commuting variety of a symmetric semisimple Lie algebra
Given an involution \(\theta \) of a finite dimensional semisimple Lie algebra \(\mathfrak g\) and the associated Cartan decomposition \(\mathfrak g=\mathfrak k\oplus \mathfrak p\), the nilpotent commuting variety of \((\mathfrak g,\theta )\) consists of pairs of nilpotent elements \((x,y)\) of \(\mathfrak p\) satisfying the condition \([x,y]=0\). An interesting structural problem concerns the questions whether this variety is equidimensional and its irreducible components are indexed by the orbits of \(\mathfrak{p}\) distinguished elements (i.e., those elements whose centralizer in \(\mathfrak{p}\) is formed by nilpotent elements). The validity of this assertion was already established by \textit{A. Premet} in the case \((\mathfrak g\times \mathfrak g,\theta )\) where \(\theta (x,y)=(y,x)\) [Invent. Math. 154, No. 3, 653--683 (2003; Zbl 1068.17006)]. The main purpose of this to prove the conjecture for other types. In all, 18 cases are covered and proven to satisfy the assertion. The proof is based largely on two technical lemmas on the orbits and a reduction principle.
scientific article; zbMATH DE number 5541197
scientific article