# Geometric Representation Theory Seminar

This page contains information about the Geometric Representation Theory Seminar at the University of Toronto. The seminar usually meets on Fridays, 3-5pm, 1015 Huron. For more information, please contact one of the organizers: Joel Kamnitzer (jkamnitz [at] math) and Sergey Arkhipov (hippie [at] math).

Previous semesters' seminars: http://www.math.toronto.edu/~jkamnitz/seminar/reptheory/index.html

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## February 26 (2010), Mikhail Mazin (Toronto)

Title: Tate's proof of the Residue Formula
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The classical Residue Formula says that for a proper algebraic curve X and a rational 1-form w the total sum of residues of w over all points of X is equal to zero. This formula can be proved in many different ways and has numerous generalizations. The most known prove is the topological one. However, it only works over complex numbers.

In the 1968, John Tate published his paper "Residues of differentials on curves", where he gave an algebraic proof of the Residue Formula. Tate expressed the residue as the trace of a certain finite-potent operator and used the properties of traces to prove the Residue Formula. The advantage of this approach is its generality. The only fact about the curve X used in the proof is that the 0-th and 1-st cohomology groups of X with coefficients in the structure sheaf are finite dimensional.

The Tate's constructions were generalized by Alexander Beilinson to the multidimensional case (A. Beilinson, "Residues and Adeles", 1980). He used it to prove the Parshin's Reciprocity Law, which is a multidimensional analog of the Residue Formula.

In this talk I will present the Tate's original constructions. However, I will try to reformulate it more in the spirit of the Beilinson's generalizations. Time permitting, I will say few words about the multidimensional case as well.

## February 12 (2010), Kiumars Kaveh (McMaster)

Title: String polytopes of Littelmann-Bernstein-Zelevinsky and Newton-Okounkov bodies
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Using algebraic/combinatorial properties of the "crystal bases", in their fundamental works Littelmann and Bernstein-Zelevinsky constructed polytopes \Delta_\lambda associated to a dominant weight \lambda of a connected reductive group G such that the integral points in Delta_\lambda are naturally in one-to-one correspondence with elements of a "crystal basis" for the irreducible representation V_\lambda. These are called the string polytopes. In this talk I discuss how the string polytopes can be defined purely geometrically in terms of the flag variety of G. That is, they naturally arise as a special case of the very general construction of the Newton-Okounkov bodies applied to the flag variety. For the most part, the talk should be accessible to anyone with only a minimum background in representation theory.

## February 5 (2010), Yiannis Sakellaridis (Toronto)

Title: Harmonic analysis on spherical varieties in terms of the wonderful compactification.
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In a recent number/representation theory seminar I described some joint results with A.Venkatesh on harmonic analysis on a spherical variety X. More precisely, the continuous spectrum of L^2(X) is understood in terms of the L^2-spectrum of normal bundles to orbits on the wonderful compactification of X. This is related to a conjectural description of the whole spectrum in terms of Arthur parameters into the "Langlands dual" group of X, which was first constructed by Gaitsgory and Nadler in the context of the geometric Langlands program. Without assuming that one was present in that talk, I will focus on explaining the main ideas behind the proofs. Also, in order to justify my presence at this seminar, I will explain a few things about the geometric structure of wonderful compactifications, their root system and the connection to the work of Gaitsgory and Nadler.

## February 2 (2010), Joseph Johns (Columbia) 4-6pm BA6183

Title: Fukaya categories of Lefschetz fibrations and quivers
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