Executive Summary : | Commuting varieties are classical objects of study in Lie theory. Historically the first result was the proof by Motzkin-Taussky in 1955 (established independently by Gerstenhaber) that the variety of pairs of commuting n x n matrices is irreducible. Richardson extended this to an arbitrary reductive Lie algebra in characteristic zero. Up to now, the properties of being Cohen-Macaulay and normal for commuting varieties are in general not verified. For a reductive Lie algebra, there is a long-standing conjecture stating that the commuting variety is always normal and reduced. Artin and Hochster claimed that the commuting variety for n x n matrices is a Cohen-Macaulay integral domain. In this proposal we will address these long-standing open questions. On the other hand, the nilpotent commuting variety for a complex semisimple Lie algebra consisting of tuples of commuting nilpotent elements, having the property that all irreducible components have the same dimension. There is not much hope for verifying Cohen-Macaulayness of nilpotent commuting varieties since their defining ideals are not radical. In this project we study the reducedness of the nilpotent commuting scheme and investigate if the defining ideal is generated by quadratic equations. For a symmetric pair, the associated commuting scheme is defined by the set of commuting tuples of elements in the -1 eigen space. The irreducibility problem for the commuting varieties associated to symmetric pairs was first considered by Panyushev and observed that the commuting variety can be reducible. He showed that for maximal rank symmetric pairs the commuting variety is irreducible, normal complete intersection and the defining ideal is generated by quadrics. In general the normality and reducedness question of the commuting varieties associated to a symmetric pair is still open and we will try to give an affirmative answer to this question. However, for a reducive Lie algebra the categorical quotient of the commuting variety by the associated reductive group behaves better. Chen and Ngo conjectured that the Chevalley restriction map from the categorical quotient scheme to the Weyl group quotient of copies of the Cartan subalgebra is an isomorphism and they proved the conjecture for the general linear group and the symplectic group. From this they concluded that the quotient scheme is reduced and normal. For the orthogonal group the question is still open and we would like to explore this question. We would also like to study the geometric properties of the quotient scheme corresponding to the symmetric pairs. So in summary, the sole motive of the proposed project is to study the algebro-geometric properties of the usual commuting scheme, the nlpotent commuting scheme and the commuting scheme associated to the classical symmetric pairs. In future, we use the obtained results to conclude the geometric properties of the Hilbert scheme of points on complex n-space. |