Prakash Belkale

The strange duality conjecture for generic curves

Let SUX(r) be the moduli space of semi-stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve X of genus g ≥ 1 over C. Recall that a vector bundle E on X is called semi-stable if for any subbundle V , deg(V )/ rk(V ) ≤ deg(E)/ rk(E). Points of SUX(r) correspond to isomorphism classes of semi-stable rank r vector bundles with trivial determinant up to an equivalence relation. For any line bundle L of degree g−1 onX define ΘL = {E ∈ SUX(r), h(E⊗L) ≥ 1}. This turns out be a non-zero Cartier divisor whose associated line bundle L = O(ΘL) does not depend upon L. It is known that L generates the Picard group of SUX(r) (for this and the precise definition of L in terms of determinant of cohomology see [DN]). Let U∗ X(k) be the moduli space of semi-stable rank k and degree k(g−1) bundles on X. Recall that on U∗ X(k) there is a canonical non-zero theta (Cartier) divisor Θk whose underlying set is {F ∈ U∗ X(k), h(X,F ) = 0}. Put M = O(Θk). Consider the natural map τk,r : SUX(r)× U∗ X(k) → U∗ X(kr) given by tensor product. From the theorem of the square, it follows that τ∗ k,rM is isomorphic to L M. The canonical element Θkr ∈ H0(U∗ X(kr),M) and the Kunneth theorem give a map well defined up to scalars:

Periods and Igusa local zeta functions

We show that the coefficients in the Laurent series of the Igusa local zeta functions I(s) = ∫ C fω are periods. This is proved by first showing the existence of functional equations for these functions. This will be used to show in a subsequent paper (by P. Brosnan) that certain numbers occurring in Feynman amplitudes (up to Gamma factors) are periods. We also give several examples of our main result, and one example showing that Euler’s constant γ is an exponential period.

Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups

In this paper we consider the eigenvalue problem, intersection theory of homogeneous spaces (in particular, the Horn problem) and the saturation problem for the symplectic and odd orthogonal groups. The classical embeddings of these groups in the special linear groups play an important role. We deduce properties for these classical groups from the known properties for the special linear groups.

Invariant theory of GL(n) and intersection theory of Grassmannians

We give an algebro-geometric relation between intersection theory of Grassmannians and invariant theory of the general linear group. From points of intersection of Schubert varieties in Gr(r,n), we produce “all” invariants in tensor products of (associated) irreducible representations of SL(r).

Vanishing and identities of conformal blocks divisors

Conformal block divisors in type A on

Transformation formulas in quantum cohomology

\bar{M}_{0,{n}}

The Tangent Space to an Enumerative Problem

are shown to satisfy new symmetries when levels and ranks are interchanged in non-standard ways. A connection with the quantum cohomology of Grassmannians reveals that these divisors vanish above the critical level.

Extremal rays in the Hermitian eigenvalue problem

The article discusses an action of the center of G on the quantum cohomology of G/Ps constructed geometrically. It is shown how to recover Bertrams Quantum Schubert Calculus from this action, and also a refinement of a formula of Fulton and Woodward for the terms of the smallest order in the product of two Schubert varieties in the Quantum cohomology of a Grassmannian. The results date to July 2001, and are needed to pose a Quantum analogue of the Horn and Saturation Conjectures.

Eigenvalue problem and a new product in cohomology of flag varieties

We will discuss recent work on the relations between the intersection theory of homogeneous spaces (and their quantum, and higher genus generalizations), invariant theory, and non-abelian theta functions. The main theme is that the analysis of transversality in enumerative problems can be viewed as a bridge from intersection theory to representation theory. Some of the new results proved using these ideas are reviewed: multiplicative generalizations of the Horn and saturation conjectures, generalizations of Fulton’s conjecture, the deformation of cohomology of homogeneous spaces, and the strange duality conjecture in the theory of vector bundles on algebraic curves.

Geometric proofs of Horn and saturation conjectures

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of