For simplicity, just let G=GL(2) or SL(2),g be the corresponding Lie algebra, K=SO(2), we have various realizations of smooth or unitary representation of G in certain function spaces. Can one give an explicit description of the corresponding (g, K)-module? (or reference is ok)

For example, let SL(2) act on unit circle, then the smooth representation of SL(2) on smooth functions of the circle has the set of trigonometric polynomials as the underlying (g, K)-module.