A quadratic inequality for sum of co-adjoint orbits

Abstract

We obtain an e¤ective lower bound on the distance of sum of coadjoint orbits from the origin. Even when the distance is zero, thus the symplectic quotient is well-de ned, our result give a nontrivial constraint on these co-adjoint orbits. In the particular case of unitary groups, we recover the quadratic inequality for eigenvalues of Hermitian matrices satisfying A+B = C: This quadratic inequality was obtained earlier by the authors using completely di¤erent means, namely Klyachko’s theory of toric stable re‡exive sheaves and the Chern number inequality for Hermitian Yang-Mills connection. 1 Introduction Given any rank r Hermitian matrix A, we may order its eigenvalues in such a way that 1 (A) 2 (A) ::: r (A) ; and denote (A) := ( 1 (A) ; ; r (A)) 2 R as its spectrum. In [K2] (Also see [Fu] for an excellent account of the subject), Klyachko discovered following series of linear inequalities for Hermitian matrices A;B;C satisfying A+B = C : X k2K k (C) X i2I i (A) + X j2J j (B) for some triple of subsets I; J;K f1; 2; :::; rg of the same cardinality and such that the associated Schubert cycle sK is a component of sI sJ . This result can be interpreted as describing the linear inequalities which determine the intersection of the sum of co-adjoint orbits O (A) + O (B) + O ( C) with the positive Weyl chamber for the unitary group, which is a convex polytope by Kirwan’s convexity theorem. Klyachko’s result was generalized to any compact Lie group by Berenstein and Sjamaar in the beautiful paper [BeS], they are 1 able to relate above eigenvalue inequalities to the convexity of the image of the moment map. All above inequalities are linear on eigenvalues of matrices. In [LW], we nd a natural quadratic inequality on these eigenvalues by relating the Hermitian matrices to stable re‡exive sheaves over the projective spaces. More precisely, we have shown Theorem 1 Suppose A1; ; AN are rank r Hermitian matrices satisfying N X