If one considers the algebra $su(2)$, it is well known that the Casimir Operator is $$ C=L_1^2+L_2^2+L_3^2. $$ It corresponds to the total angular momentum and correctly is a conserved quantity.
I would like to know which is the physical meaning of the two Casimir operators of the Poincarè algebra.
The two Casimir operators are $P^2 =P^\mu P_\mu$ where $P$ is momentum, and $W^2 =W^\mu W_\mu$ where $W$ is the so-called Pauli-Lubanski pseudovector.
Evaluating $P^2$ in a particle's rest frame, we find that $P^2 = m^2$, so the first Casimir labels representations by mass.
The Pauli-Lubanski pseudovector is a bit more complicated but also has a simple interpretation in terms of spin. By definition, it's $W^\mu = -\frac{1}{2} \epsilon^{\mu\nu\rho\sigma} P_\nu S_{\rho\sigma}$, where $S$ is the relativistic spin angular momentum. So $W^2$ has to do with spin. In particular it gives you the particle's spin for massive particles: $ W^2 = -m^2 s(s+1)$ and helicity for massless particles, $ W^2=0$ and $ W^\mu = \pm s P^\mu$.
