International Congress of Mathematicians

608 A. Klyachkospectra. By Metha-Seshadry theorem solvability of this equation is equivalent tostability inequalities (4.4). In the case under consideration holomorphic vectorbundle £ on P 1 is trivial, £ = E x P 1 , and hence its subbundle T C £ of rank pis nothing but a rational curve ip : P 1 —¥ G P (E) in Grassmannian. This allows towrite down stability condition (4.4) in terms of quantum cohomology, and eventuallyarrive at Theorem 4.1.5. Further ramificationsThe progress in Hermitian and unitary spectral problems open way for solutionof a variety of others classical, and not so classical, problems. Alost of them,however, have no holomorphic interpretation, and require different methods, borrowedfrom harmonic analysis on homogeneous spaces, symplectic geometry, andgeometric invariant theory.5.1. Multiplicative singular value problemThe problem in question is about possible singular spectrum a (AB) of productof complex matrices with given singular spectra a (A) and a(B). Recall, that singularspectrum of complex matrix A is spectrum of its radial part a (A) := A(v / A*A).For a long time it was observed that every inequality for Hermitian problem hasa multiplicative counterpart for the singular one. For example multiplicative versionof Weyl's inequality A i+i _i(A + B) < A,(A) + Xj(B) is a i+j -i(AB) < afiA)aj(B).The equivalence between these two problems was conjectured by R. C. Thompson,and first proved by the author [20] using harmonic analysis on symmetric spaces.Later on A. Alekseev, E. Alenreken, and Ch. Woodward [1] gave an elegant conceptualsolution based on Drinfeld's Poisson-Lie groups [8]. Here is a precise statementfor classical groups.Theorem 5.1. LetG be one of the classical groups SY(n,C), SO(n,C), orSp(2n,C)and L be the corresponding compact Lie algebra of traceless skew Hermitian complex,real, or quaternionic nxn matrices respectively. Then the following conditionsare equivalent(1) There exist A t £ G with given singular spectra a (Ai) = ai andAiA 2 • • • Apf = 1.(2) There exist Hi £ L with spectra A(ffj) = y^ldogCTJ andHi + H 2 + --- + H N = 0.Note, however, that neither of the above approaches solve the singular problemper se, but reduces it to Hermitian one. Both of them suggest that all three problems