Spring, 2011

• SpeakerBen Harris, MIT
  • TitleTransforms of Nilpotent Coadjoint Orbits and Wave Front Cycles of Tempered Characters
  • Time/place 1/21/2011, Friday, 11:45 am in Hill 705
  • Abstract Suppose $\pi$ is an irreducible, admissible representation of a reductive Lie group with character $\Theta_{\pi}$. By results of Barbasch-Vogan and Schmid-Vilonen, the leading term of $\Theta_{\pi}$ at one is an integral linear combination of Fourier transforms of nilpotent coadjoint orbits.

    The first half of this talk will be about understanding Fourier transforms of nilpotent coadjoint orbits. I will state the most powerful theorem in the subject due to Rossmann and Wallach. Then I will explicitly write down Fourier transforms of nilpotent coadjoint orbits for $\text{GL}(n,\mathbb{R})$.

    The second half of this talk will be about understanding which orbits occur in leading terms of characters. In particular, I will state a conjecture of David Vogan about which orbits occur in leading terms of tempered characters. If time permits, I will give some hint as to how one direction of this conjecture is proved. This will consist of giving an analogue of Kirillov's dimension formula for tempered representations of reductive Lie groups.

• Speaker Alexander Yong, University of Illinois at Urbana-Champaign
  • Title Patch Ideals and Peterson varieties
  • Time/place 3/4/2011, Friday, 11:45 am in Hill 705
  • Abstract Patch ideals encode neighbourhoods of a variety in GL_n/B. For Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen-Macaulay and Gorenstein. Consequently, we combinatorially describe the singular locus of the Peterson variety; give an explicit equivariant K-theory localization formula; and extend some results of [B. Kostant '96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties. We conjecture that the projectivized tangent cones are Cohen-Macaulay and Gorenstein, and that their h-polynomials are nonnegative and upper-semicontinuous. Similarly, we use patch ideals to briefly analyze other examples of torus invariant subvarieties of GL_n/B, including Richardson varieties and Springer fibers.

    This is based on joint work with Erik Insko (U. Iowa).

• Speaker Haisheng Li, Rutgers University at Camden
  • Title Quantum vertex algebras and their phi-coordinated modules
  • Time/place 3/11/2011, Friday, 11:45 am in Hill 705
  • Abstract I am going to talk about how to associate quantum vertex algebras in a certain sense to quantum affine algebras. First, I will give the definitions of weak quantum vertex algebras and quantum vertex algebras. Second, I will give the definition of a phi-coordinated quasi module for a weak quantum vertex algebra and give a conceptual construction. Last, I will briefly explain how one can associate weak quantum vertex algebras to quantum affine algebras.

• Speaker Vladimir Retakh, Rutgers University
  • Title A short proof of Kontsevich's cluster conjecture
  • Time/place 3/25/2011, Friday, 11:45 am in Hill 705
  • Abstract We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map K_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1}) are given by noncommutative Laurent polynomials. Joint work with A. Berenstein

• Speaker Lisa Carbone, Rutgers University
  • Title Kac-Moody groups as infinite dimensional Chevalley groups
  • Time/place 4/29/2011, Friday, 11:45 am in Hill 705
  • Abstract We describe a construction of the Tits functor for symmetrizable Kac-Moody groups G using integrable highest weight modules for the corresponding Kac-Moody algebra and a Z-form of the universal enveloping algebra. This gives a construction of Kac-Moody groups G as infinite dimensional analogs of finite dimensional Chevalley groups and naturally leads us to the Z-form G(Z) which we also construct. This is joint work with Howard Garland.