Representations of the $su(1,1)$ current algebra and probabilistic perspectives

Abstract

We construct three representations of the $su(1,1)$ current algebra: in extended Fock space, with Gamma random measures, and with negative binomial (Pascal) point processes. For the second and third representations, the lowering and neutral operators are generators of measure-valued branching processes (Dawson-Watanabe superprocesses) and spatial birth-death processes. The vacuum is the constant function $1$ and iterated application of raising operators yields Laguerre and Meixner polynomials. In addition, we prove a Baker-Campbell-Hausdorff formula and give an explicit formula for the action of unitaries $\exp( k^+(\xi) - k^-(\xi))\exp(2 \mathrm i k^0(\theta))$ on exponential vectors. We explain how the representations fit in with a general scheme proposed by Araki and with representations of the $SL(2,\mathbb{R})$ current group with Vershik, Gelfand and Graev's multiplicative measure.