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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.

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2 Answers 2

e.g. see the Wikipedia article "Representation theory of $SL_2(R)$". It has a complete description of the irreducible modules with explicit formulas for the Lie algebra action. The modules realized in functions on the circle that you mentioned are the principal series.

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See also the question Unitary representations of $SL(2, \mathbb R)$.

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