$GL^+(2,\mathbb{R})$ is the notation for positive non-zero determinant square matrices of order two.
I am reading Bump's book on Automorphic Forms and Representations. One of the major part of second chapter is about classifying admissible unitary representations of $GL^+(2,\mathbb{R})$. However, he first classifies irreducible admissible $(\mathfrak{g},K)-$ modules for $GL^+(2,\mathbb{R})$ where $K = SO(2)$.
For an admissible representation of $G$ in a Hilbert space $\mathfrak{h}$, we can form its $K$-finite vectors (basically the admissible $(\mathfrak{g},K)-$ module). Further these $K-$finite vectors have irreducible components (irreducible as a $(\mathfrak{g},K)-$ module) and we can classify these irreducible spaces based on information from operations of elements of Universal Lie algebra. Bump has classified this explicitly in later sections.
I am confused as to how does this classify admissible representations of $GL^+(2,\mathbb{R})$. In other words, I want to ask how uniquely we can identify a representation of group $G$ if we know its $K$-finite vectors. Even different representations can have isomorphic $K-$finite vectors (as given in Exercise 2.6.1 in Bump's book). So we can't say that set of $K$- finite vectors are in a way one-to-one correspondence with admissible representations.
This confusion also forced me to wonder that the problem of classification can't be explicitly stated. It seems like that classification means to give a "nice" description of something in terms of something "nice" present inside it.
UPDATE : Changed the notation for positive determinant matrices. Thanks to Paul Garett for mentioning that.