Quantum $SL_2$, Infinite curvature and Pitman's 2M-X theorem

François Chapon, Reda Chhaibi

Published 2019-04-01Version 1

It is understood that Pitman's theorem in probability theory is intimately related to the representation theory of $\mathcal{U}_{q}(\mathfrak{sl}_2)$, in the so-called crystal regime $q \rightarrow 0$. This relationship has been explored by Biane and then Biane-Bougerol-O'Connell at several levels. On the other hand, Bougerol and Jeulin showed the appearance of the Pitman transform in the infinite curvature limit $r \rightarrow \infty$ of Brownian motion on the symmetric space $SL_2(\mathbb{C})/SU_2$. In order to understanding the phenomenon, we exhibit a presentation of the Jimbo-Drinfeld quantum group $\mathcal{U}_q^\hbar(\mathfrak{sl}_2)$ which isolates the role $r$ of curvature and that of the Planck constant $\hbar$. The simple relationship between parameters is $q=e^{-r}$. The semi-classical limits $\hbar \rightarrow 0$ are the Poisson-Lie groups dual to $SL_2(\mathbb{C})$ with varying curvatures $r \in \mathbb{R}_+$. We also construct classical and quantum random walks, drawing a full picture which includes Biane's quantum walks and the construction of Bougerol-Jeulin. The curvature parameter $r$ leads to both to the crystal regime at the level of representation theory ($\hbar>0$) and to the Bougerol-Jeulin construction in the classical world ($\hbar=0$). All these results are neatly in accordance with the philosophy of Kirillov's orbit method.