# SCIENTIFIC PROGRAMS AND ACTIVITIES

April 23, 2019

## October 3-4, 2009 SOUTHERN ONTARIO GROUPS AND GEOMETRY to be held at the Fields Institute

Organizing Committee:
Joel Kamnitzer (University of Toronto)
Kevin Purbhoo (University of Waterloo)

### Speaker Abstracts

Sergey Arkhipov (University of Toronto)
Modules over the quantum group and coherent sheaves. The singular block case.

We consider a singular block in the category of modules over the Lusztig quantum group at a root of unity. The old result of Bezrukavnikov and Ginzburg identifies this category with the derived category of equivariant coherent sheaves on the Springer variety for the Langlands dual group. We outline a proof of the corresponding statement for a singular block, in terms of coherent sheaves on the cotangent bundle to a generalized flag variety for the Langlands dual group. Then we explain the relation of the statement with a possible formulation of an algebraic analog of the Freed-Hopkins-Teleman theorem.

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Chris Brav (University of Toronto)
Smaller non-commutative resolutions of rank varieties

A rank variety is the locus of matrices of some kind having rank less than or equal to some fixed non-negative integer. Typically,
rank varieties are singular and have standard resolutions given by total spaces of vector bundles over Grassmannians. When such resolutions are small, the cohomology of the resolution is isomorphic to the intersection cohomology of the rank variety.
In the case of a skew-symmetric rank variety, the standard resolution fails to be small and we show how to replace it with a smaller non-commutative resolution whose Grothendieck group is isomorphic to the intersection cohomology of the rank variety.

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Allen Knutson (Cornell University)
Frobenius splitting and juggling patterns

Given a reducible hypersurface f=0 in affine n-space, we can construct many other subschemes: decompose the hypersurface, intersect the pieces, decompose the intersections, repeat.

Theorem: if f is degree n, and the number of Z/p solutions
is not a multiple of p, then all these subschemes are reduced. (The same follows in characteristic 0, if it holds for infinitely many p.) This is true if f's leading term is the product of all the variables.

I'll prove this, and describe a remarkable f on the space of matrices (or really, the Grassmannian), whose associated strata are indexed by juggling patterns. This part is joint with Thomas Lam and David Speyer.
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Mikhail Mazin (University of Toronto)

Geometric Theory of Parshin's Residues

In the early 70-ties Parshin introduced his notion of the multidimensional residue, which is a generalization of the classical one-dimensional residue. The main difference between the Parshin's definition and the one-dimensional case is that in higher dimensions one computes the residue not at a point but at a complete flag of subvarieties X = Xn ¾ ¢ ¢ ¢ ¾ X0; dimXk = k: Parshin also proved a Reciprocity Law for residues: if one fixes all elements of the flag, except Xk; where 0 < k < n; and consider all possible choices of Xk; then only finitely many of these choices give non-zero residues, and the sum of these residues is zero.
Parshin's constructions are completely algebraic. In fact, they work in very general settings, not only over complex numbers. However, in the complex case one would expect a more geometric variant of the theory. In this talk I will present my current work under Prof. Khovanskii supervision on the geometric theory of Parshin's residues. There are two parts of the work: in the first part we use the geometry of Stratied Spaces to construct smooth cycles, such that the residues are the integrals over them, and to prove the Parshin's Reciprocity Law for residues. In the second part we use the Resolution of Singularities technics to study the geometry of a singular variety near a complete flag of subvarieties.
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Nicholas Proudfoot (University of Oregon)
Goresky-MacPherson duality and deformations of Koszul algebras

Goresky and MacPherson made the following cute observation: consider a Type A Springer variety along with its natural torus action, and look at Spec of its equivariant cohomology ring, which consists of a union of linear subspaces of a vector space. Now consider the collection of subspaces of the dual vector space obtained by taking the perps of all of these guys. It turns out that this is itself the spectrum of the equivariant cohomology ring of a certain partial flag variety!

I will explain some of the beautiful structures that are lurking behind this picture, and in so doing generate many more analogous examples, both known and conjectural.

This is joint work with the Symplectic Duality Crew (Tom Braden, Tony Licata, and Ben Webster) and Chris Phan.
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Michael Roth (Queens University)
Cup product of line bundles on complete flag varieties

Given a semisimple algebraic group G (like SL_{n+1}) the complete flag variety X=G/B is an algebraic variety which is a geometric incarnation of the Lie theory of G. In particular, the cohomology of line bundles on X produce all the irreducible representations of G (in more than one way). This talk will give answers to the following two natural questions, which mix representation theory and geometry:

Q1: Given line bundles L_1 and L_2 on X, with nonzero cohomology in degrees d_1 and d_2, when is the cup product map

H^{d_1}(X,L_1) \otimes H^{d_2}(X, L_2) -----> H^{d}(X, L)

surjective (where d=d_1+d_2 and L=L_1\otimes L_2)?

Q2: Fixing two irreducible representations V_1 and V_2 of G, what irreducible subrepresentations of the tensor product V_1\otimes V_2 can be realized via the cup product map above?
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Alistair Savage (University of Ottawa)
Equivariant map algebras

Suppose a finite group acts on a scheme (or algebraic variety) X and a finite-dimensional Lie algebra g.  Then the space of equivariant algebraic maps from X to g is a Lie algebra under pointwise multiplication.  Examples of such equivariant map algebras include (multi)current algebras, (multi)loop algebras, three point Lie algebras, and the (generalized) Onsager algebra.  In this talk we will present a classification of the irreducible finite-dimensional representations of an arbitrary equivariant map algebra.  It turns out that (almost) all irreducible finite-dimensional representations are evaluation representations.  As a corollary, we recover known results on the representation theory of particular equivariant map algebras (for instance, the loop algebras and the Onsager algebra) as well as previously unknown classifications of other equivariant map algebras (for example, the generalized Onsager algebra).  All such classifications are specializations of the general theorem.  This is joint work with Erhard Neher and Prasad Senesi.
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Michael Wong (McGill University)
Wonderful Compactifications and Moduli of Principal Bundles

Starting with a fixed vector bundle W over a Riemann surface, it is possible to construct families of vector bundles that occur as full rank subsheaves of W that give open sets of the moduli space of stable vector bundles. Essentially, the family is obtained by twisting the transition function for W in punctured neighbourhoods of a finite set of points on the Riemann surface. We may ask whether such a construction can be mimicked for an arbitrary principal bundle. In the case that the structure group is a semisimple algebraic group of adjoint type, we may use the wonderful (De Concini-Procesi) compactification to produce analogous "twists," or Hecke modifications, supported at points. In some cases, these modifications can indeed be put together to give parametrizations of the moduli space of principal bundles. If time permits, we will also look at the Hitchin-type space of pairs of bundles and connections as seen from the perspective of this parametrization.