Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras

We introduce a generalization of the classical Hall-Littlewood and Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove that these Kostka-Foulkes polynomials coincide with the natural generalization of Lusztig's $t$-analog of weight multiplicities, thereby extending a theorem of Kato. For $g$ an affine Kac-Moody algebra, we define $t$-analogs of string functions and use Cherednik's constant term identities to derive explicit product expressions for them.Comment: 19 page

Navier-Stokes solver using Green's functions II: spectral integration of channel flow and plane Couette flow

The Kleiser-Schumann algorithm has been widely used for the direct numerical simulation of turbulence in rectangular geometries. At the heart of the algorithm is the solution of linear systems which are tridiagonal except for one row. This note shows how to solve the Kleiser-Schumann problem using perfectly triangular matrices. An advantage is the ability to use functions in the LAPACK library. The method is used to simulate turbulence in channel flow at $Re=80,000$ (and $Re_{\tau}=2400$) using $10^{9}$ grid points. An assessment of the length of time necessary to eliminate transient effects in the initial state is included

A note on exponents vs root heights for complex simple Lie algebras

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra, the partition formed by its exponents is dual to that formed by the numbers of positive roots at each height.Comment: 5 page