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In the first part of the seminar I will give a short review of the frame-independent formulation of Newtonian gravity, called Newton-Cartan Gravity, and explain why there is a renewed interest into non-relativistic gravity in general.  In the second part I will discuss, as a particular application,  a recent proposal for an Effective Field Theory describing a massive spin-2 mode (the so-called GMP mode) in the Fractional Quantum Hall Effect.

This project examines the structure of a certain category of representations of a Lie algebra called Whittaker modules. Whittaker modules generalize highest weight modules, and the structure of the category is similar to that of Bernstein-Gelfand-Gelfand’s category O. In particular, Whittaker modules have finite length composition series and all irreducible Whittaker modules appear as quotients of standard Whittaker modules, which are generalizations of Verma modules. Using the localization theory of Beilinson-Bernstein, one obtains a beautiful geometric description of Whittaker modules as twisted sheaves of D-modules on the associated flag variety. I use this geometric setting to develop an algorithm for computing the multiplicities of irreducible Whittaker modules in the composition series of standard Whittaker modules. This algorithm establishes that the multiplicities are determined by parabolic Kazhdan-Lusztig polynomials.