Related papers: Convolution for quiver varieties via cup product o…

In this paper we use the gradient flow equation introduced in [10] to construct a Morse complex for the Hamiltonian action $\mathbb A_H$ on a mixed regularity space of loops in the cotangent bundle $T^*M$ of a closed manifold $M$.…

The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant over a complex curve carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion…

We present simple conditions which guarantee a geometric convolution algebra to behave like a variant of the quasi-hereditary algebra. In particular, standard modules of the affine Hecke algebras of type $\mathsf{BC}$, and the quiver Schur…

Our main theorem describes the degree 0 cohomology of Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod p Hodge-type Shimura varieties with hyperspecial level at p, mirroring the well known analogue…

For algebraic varieties defined by hyperkahler or, more generally, algebraic symplectic reduction, it is a long-standing question whether the "hyperkahler Kirwan map" on cohomology is surjective. We resolve this question in the affirmative…

We construct a $(\mathfrak{gl}_2, B(\mathbb{Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb{P}^1$, landing in the compactly supported completed…

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of…

We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the…

Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of a particular form, then $F(s)=L_f(s)$ for some…

The Kac-Moody affine Hecke algebra $\mathcal{H}$ was first constructed as the Iwahori-Hecke algebra of a $p$-adic Kac-Moody group by work of Braverman, Kazhdan, and Patnaik, and by work of Bardy-Panse, Gaussent, and Rousseau. Since…

Let T be a compact torus and X a nice compact T-space (say a manifold or variety). We introduce a functor assigning to X a "GKM-sheaf" F_X over a "GKM-hypergraph" G_X. Under the condition that X is equivariantly formal, the ring of global…

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a Hermitian-Einstein metric if and only if the bundle…

We give a `geometrical' construction of an action of a Heisenberg algebra on the homology of the moduli spaces of torsion free sheaves on a complex smooth connected projective surface, framed along a smooth connected genus zero curve. This…

We establish a correspondence between vector-valued modular forms with respect to a symmetric tensor representation and quasimodular forms. This is carried out by first obtaining an explicit isomorphism between the space of vector-valued…

We extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines. We introduce a set of invariants for a vector bundle on the product of two projective lines,…

We introduce a new version $kk^{\rm alg}$ of bivariant $K$-theory that is defined on the category of all locally convex algebras. A motivating example is the Weyl algebra $W$, i.e. the algebra generated by two elements satisfying the…

We use operator algebras and operator theory to obtain new result concerning Berezin quantization of compact K\"ahler manifolds. Our main tool is the notion of subproduct systems of finite-dimensional Hilbert spaces, which enables all…

We show that the higher-dimensional Heegaard Floer homology between tuples of cotangent fibers and the conormal bundle of a homotopically nontrivial simple closed curve on $T^2$ recovers the polynomial representation of double affine Hecke…

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in…

We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This…