Illustrating Delorme’s intertwining conditions on SL(2,ℝ) and beyond
Appears in collection : 2025 - T1 - WS1 - Intertwining operators and geometry
The Paley-Wiener space for compactly supported smooth functions $C^\infty_c(G)$ on a semisimple Lie group $G$ is characterised by certain intertwining conditions, known as \textit{Delorme's intertwining conditions}, which are challenging to work with. Using the concept of Collingwood's boxes, we demonstrate how these relationships can be simplified and visualised in specific cases such as $G = \mathrm{SL}(2,\mathbb{R})$, its finite products, and $\mathrm{SL}(2,\mathbb{C})$. Additionally, we explore how this criterion for the Paley-Wiener space can be applied to analyse the solvability of invariant differential operators acting between sections of homogeneous vector bundles over the corresponding symmetric spaces.
