2012–2013 Abstracts
Fall 2012
7 September 2012
- Nicholas Proudfoot, University of Oregon
- Quantizations of conical symplectic resolutions
- The most studied example of a conical symplectic resolution is the cotangent bundle $M$ of the flag manifold $G/B$, which resolves the nilpotent cone in $\text{Lie}(G)$. Much of what goes under the name "geometric representation theory" is the study of this resolution. Here are two cool features of this subject:
- –If you construct a deformation quantization of M and take global sections, you get the ring of global (twisted) differential operators on the flag variety, which is isomorphic to a central quotient of the universal enveloping algebra of $\text{Lie}(G)$. This allows you to study representations of $\text{Lie}(G)$ in terms of sheaves on $M$.
- –There is a natural action of "convolution operators" on the cohomology of $M$ which provides a geometric construction of the regular representation of the Weyl group of $G$. This action can be promoted to a braid group action on a category by replacing cohomology classes with sheaves.
- I will make the case that these two phenomena fit neatly into a theory that applies to arbitrary conical symplectic resolutions, including (for example) quiver varieties, hypertoric varieties, and Hilbert schemes of points on ALE spaces.
- This is joint work with Braden, Licata, and Webster.
14 September 2012
- Balazs Elek, Cornell University
- Computing the standard Poisson structure on Bott-Samelson varieties
- Bott-Samelson varieties were first introduced as resolutions of singularities of Schubert varieties but they also have interesting geometric properties of their own. They admit naturally defined affine coordinate charts which allow the computation of geometric quantities in coordinates.
- Poisson Geometry is widely used in the study of quantum groups and quantization. A holomorphic Poisson structure $\Pi$ on Bott-Samelson varieties associated to a complex semisimple Lie group $G$ was defined and studied by J.-H. Lu. In particular, Lu showed that $\Pi$ was algebraic and gave rise to an iterated Poisson polynomial algebra associated to each affine chart of the Bott-Samelson variety.
- In this talk, I will introduce $\Pi$ on Bott-Samelson varieties, and compute it in the affine charts, describing the obstacles and the tools used in the proof. Surprising side-products of the proof are that $\Pi$ depends on $\text{Lie}(G)$ and the choice of a basis only, and that if a Chevalley basis is chosen then $\Pi$, in any of the natural affine charts, is defined over the integers.
- This is the topic of my M. Phil. thesis written at The University of Hong Kong.
21 September 2012
- Andrei Rapinchuk, University of Virginia
- Weakly commensurable groups, with applications to differential geometry
- The notion of weak commensurability was introduced in ongoing joint work with Gopal Prasad on length-commensurable and isospectral locally symmetric spaces. We have been able to determine when two arithmetic subgroups are weakly commensurable. This leads to various geometric results, some of which are related to the famous question "Can one hear the shape of a drum?"
28 September 2012
- Dan Barbasch, Cornell University
- Hermitian Forms for Iwahori-Hecke Algebras
- This talk will discuss star operations for Iwahori-Hecke algebras, joint work with Dan Ciubotaru. Hecke algebras are structures which are used to study the representation theory of $p$-adic groups. In particular by results of Barbasch-Moy and subsequently Barbasch-Ciubotaru, there is a precise relation between the unitary dual of a block of representations of a $p$-adic group and a particular Iwahori-Hecke algebra. In order to talk about unitarity for an algebra, one needs a star operation. For semisimple Lie algebras, star operations are essentially parametrized by real forms. An analogous situation exists for Hecke algebras, but the situation is more rigid. This work is modeled after results in the real case by Adams-Trapa-Yee-van Leeuwen and Vogan.
12 October 2012
- Martin Kassabov, Cornell University
- Word values in $p$-adic and adelic groups
- (Joint work with N. Avni, T. Gelander and A. Shalev.) A word $w$ is an element of a free group $F_d$ on the free generators $x_1$,$\dots$, $x_d$. Given such a word $w$ and a group $G$ we define a word map from $G^d$ to $G$ induced by substitution. The image of this map is denoted by $w(G)$. Typically for finite groups $G$ the set $w(G)$ is very large. For example recent works of Shalev and Larsen show that for a fixed word $w$ and large enough finite simple group $G$, every element of $G$ is a product of two values of $w$, namely $w(G)^2=G$. I will explain how this result extends to some infinite groups, namely groups of Lie type over $p$-adic numbers and adèles.
19 October 2012
- Victor Protsak, SUNY Oswego
- Hidden symmetries in $GL_n$ to $GL_{n-2}$ branching
- When an irreducible representations of a group is restricted to a subgroup, it usually decomposes into several pieces. This decomposition, known as branching, is rarely multiplicity-free. For reductive algebraic groups, the corresponding multiplicity spaces fit together into a certain commutative algebra, the branching algebra. Branching from $GL_n$ to $GL_{n-1}$ is multiplicity-free and well-understood; it is the first step in the construction of Gelfand-Tsetlin bases. By contrast, branching from $GL_n$ to $GL_{n-2}$ is related to symplectic branching and somewhat mysterious. In this case, there is a canonical action of $GL_2$ on the branching multiplicity spaces, however, they are not multiplicity-free even as $GL_2$-modules. Recently, Wallach and Yacobi discovered extra ``hidden symmetries'': there exists a compatible action of an $(n-1)$-fold product of $\mathfrak{sl}_2$'s that resolves the multiplicities. Kim and Yacobi then showed how this results in a standard monomial basis in the branching algebra. The origin of the hidden symmetries remains unclear. In an ongoing project with Sangjib Kim, we realize them by explicit differential operators acting on the branching algebra. Two interesting features of this situation are that the action is patched together from different pieces of the branching algebra and that our expressions for differential operators are rational, i.e. involve denominators.
26 October 2012
- Ka Yue Wong, Cornell University
- The Kraft-Procesi Model and K-type Multiplicities of Nilpotent Orbit Closures
- Let $G$ be a Lie group. An element $x$ in the Lie algebra of $G$ is called nilpotent if $(\text{ad}x)^n=0$ for some large $n$. A nilpotent orbit is the set of $G$-conjugates of a nilpotent element.
- For any complex classical Lie group $G$, Kraft and Procesi constructed the ring of regular functions on the Zariski closure of any nilpotent orbit. It was used to determine whether the orbit closure is a normal variety or not. In 2003, R. Brylinski constructed a Dixmier algebra of the orbit closure, based on the Kraft-Procesi construction. In this talk, I will cover some representation theoretic aspects of the Brylinski model. In particular, if the ring of regular functions on the orbit and its Zariski closure are considered as a $G$-representations, I will give the algorithms computing the multiplicities of some small irreducible $G$-representations on both rings.
16 November 2012
- Tara Holm, Cornell University
- The Morse-Bott-Kirwan condition is local
- Kirwan identified a condition on a smooth function under which the usual techniques of Morse-Bott theory can be applied to this function. In joint work with Yael Karshon, I have proved that if a function satisfies this condition locally, then it also satisfies the condition globally. As an application, we can use the local normal form theorem for a Hamiltonian Lie group action on a symplectic manifold to recover Kirwan's result that the norm square of a momentum map satisfies Kirwan's condition. I will give an outline of the proof of the main theorem, and show how this simplifies Kirwan's result on the norm square of the momentum map. I will not assume any background in symplectic geometry.
Spring 2013
25 January 2013
- Xuhua He, Hong Kong University of Science and Technology
- Affine Weyl groups, affine Hecke algebras and affine Deligne-Lusztig varieties
- Using some nice combinatorial properties on affine Weyl groups, we establish a relation between representations of affine Hecke algebras and structure of affine Deligne-Lusztig varieties. More precisely, we have the $\text{degree}=\text{dimension}$ theorem, which relates the degree of class polynomials of affine Hecke algebras and the dimension of affine Deligne-Lusztig varieties. As a consequence, we verify the Gortz-Haines-Kottwitz-Reuman conjecture on dimension of affine Deligne-Lusztig varieties.
15 February 2013
- Omer Offen, Technion–Israel Institute of Technology
- On the distinguished automorphic spectrum for $(Sp(2n),Sp(n)\times Sp(n))$
- We introduce the notion of the automorphic spectrum of a group $G$ distinguished by a subgroup $H$ and discuss cases where we can provide some information on the distinguished space. For $(GL(2n),Sp(n))$ we describe the distinguished spectrum completely by recent results of Sunshuke Yamana. We further discuss joint work with Erez Lapid on the case $(Sp(2n),Sp(n)\times Sp(n))$ where we provide a partial description of the distinguished spectrum and its relation with the descent construction of Ginzburg-Rallis-Soudry.
21 February 2013
- Peter Trapa, University of Utah
- Unitary representations of reductive Lie groups
- Unitary representations of Lie groups appear everywhere in mathematics: in harmonic analysis (as generalizations of the sines and cosines appearing in classical Fourier analysis); in number theory (as spaces of modular and automorphic forms); in quantum mechanics (as "quantizations" of classical mechanical systems); and in many other places. They have been the subject of intense study for decades, but their classification has only recently emerged in a preprint authored jointly with Adams, van Leeuwen, and Vogan. Perhaps surprisingly, the classification has inspired connections with interesting geometric objects (equivariant mixed Hodge modules on flag varieties).
- The purpose of this talk is to review the history and motivation behind the study of unitary representations, offer a few hints about algebraic and geometric approaches to them, and explain the key new idea which makes their classification possible. In retrospect, the new idea has the famous "unitary trick" of Hermann Weyl as its antecedent.
22 February 2013
- Peter Trapa, University of Utah
- Computing invariant forms on Harish-Chandra modules
- I will explain how to compute signatures of invariant forms on irreducible Harish-Chandra modules. This provides a means to compute the unitary dual of any fixed reductive group $G$ (as discussed in the Oliver Club on Thursday) and is joint work with Adams, van Leeuwen, and Vogan.
1 March 2013
- Birgit Speh, Cornell University
- Branching laws for infinite dimensional representations: An example
- I will discuss the restriction of some induced representations of the orthogonal group $O(n,1)$ to the subgroup $O(n-1,1)$. The representations of $O(n,1)$ are infinite dimensional, usually not unitary and they do not decompose into a direct sum of irreducible representation of $O(n-1,1)$. To study the restriction new families of intertwining operators are constructed using analytic tools. This is joint work with T. Kobayashi.
14–15 March 2013
- Ivan Cherednik, University of North Carolina
- New theory of $q,t$-hypergeometric and $q$-Whittaker functions I, II
- The lectures will be devoted to the new theory of global difference hypergeometric functions, one of the major applications of the double affine Hecke algebras and a breakthrough in the classical harmonic analysis on symmetric spaces. They were introduced in 1996, 150 years after Heine's definition, which remained unchallenged since then. The analytic theory of these functions and their $q$-Whittaker limits was completed only recently (the speaker and Jasper Stokman). If time allows, connections with Gromov-Witten invariants of flag varieties and the algebraic theory of affine flag varieties will be discussed.
29 March 2013
- Ana Pires, Cornell University
- The topology of toric origami manifolds
- A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we will examine the toric origami case: we will recall how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds. These results are from ongoing joint work with Tara Holm.
12 April 2013
- Chi-Kwong Fok, Cornell University
- Real K-theory of compact simply-connected Lie groups
- KR-theory, which was introduced by Atiyah, is a version of topological K-theory for the category of topological spaces equipped with an involutive homeomorphism. Seymour obtained the $KR^*(\text{pt})$-module structure of $KR^*(G)$, where $G$ is a compact, connected and simply-connected Lie group equipped with a Lie group involution, but was unable to compute the ring structure. In this talk we will first review Hodgkin's result on complex K-theory of $G$, which is a certain exterior algebra, and Seymour's results. Then we will give a description of the ring structure of $KR^*(G)$ which, as it turns out, is not an exterior algebra in general. Time permitting, we will discuss some partial results on the equivariant KR-theory case.
18 April 2013
- Peter Sarnak, Princeton University and Institute for Advanced Study
- Quantum variance for the modular quotient
- We compute the variance of an observable $f$ measured on the eigenfunctions of the Laplacian on the modular surface. The result agrees with the classical variance after correcting by a subtle arithmetic factor—namely the central value of the corresponding L-function. Joint work with P. Zhao.
26 April 2013
- Andrew Sale, Cornell University
- A Geometric Version of the Conjugacy Problem in Semisimple Lie Groups
- For a finitely generated group, the conjugacy problem asks if there is an algorithm which determines whether a pair of elements are conjugate or not. This is one of Max Dehn's well-known decision problems and has been widely studied over the past century. Related to this problem is the question of understanding, for a pair of conjugate elements, the family of conjugators between them. We focus in particular on estimating the minimal length of such an element. This problem naturally extends beyond the reach of the conjugacy problem itself and can be asked of any group which admits a left-invariant metric. We will discuss a result for pairs of hyperbolic elements in a semisimple Lie group. The issue of extending results from here to their lattices, such as $SL(n,\mathbf{Z})$, remains unsolved, with questions raised of a number theoretic nature.
3 May 2013
- Bruce Fontaine, Cornell University
- Rotation of tensors and geometric representation theory
- The study of the invariant space of a tensor product of representations is classical. We will describe two geometric bases for this invariant space. One will be 'preserved' by rotation of tensor factors and the other will diagonalize it. By doing so we will solve a conjecture concerning the cyclic sieving phenomenon for Young tableaux and generalize the result to other crystals. This is joint work with Joel Kamnitzer.