Since so(2,1) is a noncompact Lie algebra, its unitary representations are all infinite-dimensional. Now, I have seen that the unitary representations of so(2,1) can be divided into three categories:
- Discrete series - with a lowest or highest state satisfying $J_{\pm}|j,m\rangle = 0$, where $C|j,m\rangle = -j(j+1) |j,m\rangle$ ($C$ is the Casimir operator) and $J_3|j,m\rangle = m|j,m\rangle$. Also $J_\pm = J_1 \pm iJ_2$.
- Principal series.
- Complementary series.
For principal and complementary series, there is no lower or upper bound for the $m$ values. Now, I have seen in most books and literature that for the discrete series, $j$ only takes integer or half-integer values rather than continuum values which gives the name discrete series. However, I cannot find any reason why should $j,m$ only can have integer or half-integer values if there is a lower bound on the $m$ values.
In other words, starting with the assumption that there exists a lower or upper bound of the value $m$, how can we conclude that $j$ has to be discrete?