Weight-Lattice Discretization of Weyl-Orbit Functions

摘要

Weyl-orbit functions have been defined for each simple Lie algebra, andpermit Fourier-like analysis on the fundamental region of the correspondingaffine Weyl group. They have also been discretized, using a refinement of thecoweight lattice, so that digitized data on the fundamental region can beFourier-analyzed. The discretized orbit function has arguments that areredundant if related by the affine Weyl group, while its labels, the Weyl-orbitrepresentatives, invoke the dual affine Weyl group. Here we discretize theorbit functions in a novel way, by using the weight lattice. A cleaner theoryresults, with symmetry between the arguments and labels of the discretizedorbit functions. Orthogonality of the new discretized orbit functions isproved, and leads to the construction of unitary, symmetric matrices withWeyl-orbit-valued elements. For one type of orbit function, the matrixcoincides with the Kac-Peterson modular $S$ matrix, important forWess-Zumino-Novikov-Witten conformal field theory.