Quantum Toda chain revisited

I am planning to give an account of some of my  results which describe  

a construction of  the eigenfunction of the Toda chain in terms of representation theory. More exactly,

the construction of an explicit integral representation written in special (separated) coordinates. 

The construction uses  novel realisation of the principal series representations. 

I will put into the same framework the above mention  construction of the Toda 

chain wave  function and  the Vortex counting function coinciding

with a certain limit of the instanton   counting function.

My talk is based on joint works with A.Gerasimov and S. Oblezin.