For generators of the Lorentz group ($\hat {R}_{k}$ corresponds to the generators of 3-rotations, $\hat {L}_{k}$ corresponds to the generators of the boosts) we have the following algebra: $$ [\hat {R}_{i}, \hat {R}_{j} ] = -\varepsilon_{ijk}\hat {R}_{k}, \quad [\hat {R}_{i}, \hat {L}_{j} ] = -\varepsilon_{ijk}\hat {L}_{k}, \quad [\hat {L}_{i}, \hat {L}_{j} ] = \varepsilon_{ijk}\hat {R}_{k}. $$ For the splitting of algebra, we can introduce operators $$ \hat {J}_{k} = \hat {R}_{k} + i\hat {L}_{k}, \quad \hat {K}_{k} = \hat {R}_{k} - i\hat {L}_{k}. $$ So $$ [\hat {J}_{i}, \hat {J}_{j} ] = -\varepsilon_{ijk}\hat {J}_{k}, \quad [\hat {K}_{i}, \hat {K}_{j} ] = -\varepsilon_{ijk}\hat {K}_{k}, \quad [\hat {J}_{i}, \hat {K}_{j}] = 0. $$ So, each irreducible representation of Lie algebra is characterized by $(j_{1}, j_{2})$, where $j_{1}$ is max eigenvalue of $\hat {J}_{3}$ and $j_{2}$ is max eigenvalue of $\hat {K}_{3}$.
Then I can classify objects that transform through the matrices of the irreducible representations of the operators algebra, $$ \Psi_{\mu \nu}' = S^{j_{2}}_{\mu \alpha }S^{j_{2}}_{\nu \beta}\Psi_{\alpha \beta}, $$ where $S^{j_{i}}_{\gamma \delta}: (2j_{i} + 1)\times (2j_{i} + 1)$.
Also I know, that sum $j_{1} + j_{2}$ corresponds to spin.
So, the questions.
How can we construct Hermitian operator which has max eigenvalue $j_{1} + j_{2}$ (the sum $j_{1} + j_{2}$ is connected with spin, so it must be correspond to some Hermitian operator)?
How exactly this operator connected with irreducible representation of 3-rotation group (like as it is an irreducible representation of generator of rotations: $j_{1} + j_{2} \Rightarrow \frac{1}{2}\hat {J}_{k} + \frac{1}{2}\hat {K}_{k} = \hat {R}_{k}$)?