In this talk, we plan to explore the natural question of how to compare different $G$-continuous norms on a Harish-Chandra module with a fixed growth rate. We demonstrate that the space of equivalence classes of these norms features both maximal and minimal elements, as well as a natural Sobolev distance. This leads to a new invariant, which we term the “Sobolev gap.” We will also discuss the natural question of estimating the Sobolev gap uniformly over all irreducible Harish-Chandra modules, offering a quantitative version of the Casselman-Wallach globalization theorem. Additionally, we provide explicit values for the Sobolev gap in the case of Harish-Chandra modules for $SL(2,\mathbb{R})$.

The video of this talk is available on the IISc Math Department channel.