Reposting my question from math.stackexchange: What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory (http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf):
The subject of "geometric representation theory" relates Verma modules to the geometry of the flag variety $G/B$. $B$ acts on the left, with a finite number $\lvert W \rvert$ of $B$-orbits (this is the Bruhat decomposition). For $G = SL(2;\mathbb C)$, $G/B = \mathbb CP_1$ and there are two $B$-orbits: the Riemann sphere minus the South pole, and the South pole, of dimension 2 and 0 respectively. Verma modules correspond to delta-function distributions on the $B$-orbits, dual Verma modules to holomorphic functions on the orbits, singular at the boundary.
Is there an analogue of meromorphic functions in this setting? Nevanlinna theory, perhaps? (I ask of Nevanlinna theory to, perhaps, make connections with Vojta's dictionary of some sort)