International Congress of Mathematicians

3.4. Components of tensor productSpectral Problems 605In the previous section we explain that stability inequalities (3.5) (44> (UK))via toric Donaldson-Yau theorem solve Hermitian spectral problem. To relate thiswith tensor product part of Theorem 2.1 we need another interpretation of thestability inequalities via Geometric Invariant Theory [26].Recall, that point x £ P(V) is said to be GIT stable with respect to linearaction G : V if G-orbit of the corresponding vector x £ V is closed and its stabilizeris finite. LetX = T a xT ß x T 1be product of three flag varieties of the same types as flags of the filtrationsE a ,E ß ,E~>, and £ a be line bundle on the flag variety T a induced by characteriv a : diagli,x 2 ,... ,x n ) >-¥ x^1 x^2 • • • x\ n ,where a : cti > 02 > • • • > a n is the spectrum of filtration E a , i.e. spectrum of thecorresponding operator H a .Observation 3.4. Vector bundle £ = £(E a ,E ß ,triplet of flagsE 1 ) is stable iff the correspondingi = F a xP 3 xF 1 ÉFxf' î xP = l 4 P(F(X, Cj)is a GIT stable point w.r. to group SY(E) and polarization £ = £ a M £ ß M C 1 .This observation is essentially due to Alumford [25]. Notice that by Borei-Weil-Bott theorem [5] the space of global sections F(J 7a ,£ a ) = V a is just an irreduciblerepresentation of SL(£') with highest weight a. Hence Y(X,£j) = V a ® \fi ® \fi.Every stable vector x can be separated from zero by a G-invariant section of £ N .Therefore triplet of flags in generic position is stable iff [VJV Q ® Yfiß ® VJV 7 ] SL(^ 7^ 0for some N > 1. This proves the last part of Theorem 2.1, modulo the saturationconjecture.4. Unitary operators and parabolic bundlesWe have seen in the previous section that solution of the Hermitian spectralproblem amounts to stability condition for toric bundles. A remarkable ramificationof this idea was discovered by S. Angihotri and Ch. Woodward [2] for unitaryspectralproblem.Let U £ SU(n) be unitary matrix with unitary spectrumLet's normalize exponents A, as followse(U) = (e 2 " Al ,e 2 " A V-- ,e 2 " A ").' Ai > A 2 > • • ^ X n ,+ x n =X(U) := t Ai + A 2 + --- • • + A„ = 0, (4.1)^ Ai — A„ A„ < < 1, 1,and, admitting an abuse of language, call X(U) spectrum of U.