This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras:
37602. Like some others who started graduate study in the 1960s with almost no physics background but with an interest in abstract mathematics, I was drawn to algebraic Lie theory for mainly esthetic reasons. I also had no background in differential geometry or Lie groups. So when I bought a copy of Jacobson's newly-published book on Lie algebras at a bookstore in Ithaca I had no appreciation of the historical connections of the subject.
The eminent Dutch physicist H.B.G. Casimir was apparently the first to introduce an explicit second degree invariant (unique up to scalars) in the center of $U(\mathfrak{g})$, now called the Casimir element or Casimir invariant.
Roughly speaking, this involves fixing a basis of $\mathfrak{g}$ along with its dual basis under the Killing form, then adding the respective products. Sometimes it is convenient to recast the answer in terms of PBW monomials for the given ordered basis.
On the mathematical side, Chevalley and Harish-Chandra determined the full center of $U(\mathfrak{g})$: it is a polynomial algebra in $\ell$(= rank of $\mathfrak{g}$) variables. Generators can be taken to be homogeneous of uniquely determined degrees. Moreover, the center is isomorphic in a natural way (but requiring a subtle $\rho$-twist) to the algebra $U(\mathfrak{h})^W$ of invariants of the Weyl group relative to a fixed Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. The earlier MO question involved this algebra and its (non-unique) homogeneous generators of degrees $2 = d_1, d_2, \ldots, d_\ell$. Key papers were those by Harish-Chandra (Trans. AMS 1951) and Chevalley (Amer. J. Math. 1955), the latter generalizing the fundamental theorem on elementary symmetric polynomials for $W=S_n$.
My question then is:
What role did Casimir's work play in this mathematical development?
A related matter is the practice of referring to homogeneous generators of the center of the enveloping algebra as "Casimir operators": how far was Casimir himself involved in this direction beyond his degree 2 invariant?
ADDED: The short article referenced by mathphysicist is illuminating and may be the
best published indication of Casimir's influence on subsequent representation theory. I was at first hoping to find a more definite paper trail, but this may not exist and would probably reach back before Math Reviews. What struck me most in browsing through the first volume of Harish-Chandra's collected papers (1944-54) was the abrupt transition around 1948 from his work in physics like Motion of an electron to mathematics like Faithful representations of Lie algebras and of course his foundational 1951 paper I mentioned. Nowhere along the way do I see any direct citation of Casimir's papers, though the 1950 paper Lie algebras and the Tannaka duality theorem does quote the "Casimir operator" in rank 1 as well known and uses it as a stepping-stone to the general case. Since Harish-Chandra studied physics with Dirac in his early years, it was probably he who imported Casimir's idea into representation theory. But Chevalley was at the time also a major influence on Harish-Chandra's thinking, so it's all hard to document. (They both taught at Columbia for some time.)
Best Answer
At the first glance it appears that he more or less just gave the first nontrivial example(s) of what was later called the Casimir operators.
His obituary says:
On 1 May 1931 he wrote a letter from The Hague to the famous Gottingen mathematician Hermann Weyl and announced: ‘While studying the quantum-mechanical properties of the asymmetic rotator I arrived at some ‘results’ (?) concerning the representation of continuous groups.’ He then sketched his findings on the matrix elements of the irreducible representations for the three-dimensional rotation group, and a possible extension for semi-simple groups in general, where he introduced what was later called the ‘Casimir operator’. This operator turned out to be a multiple of the unit-operator and may be used to characterize in an elegant way the irreducible representations of a given continuous group. To Casimir’s question, ‘Whether the case is worth considering?’, Weyl answered definitely ‘Yes’. Hence the Leiden doctoral candidate published his mathematical results in a paper, communicated by Ehrenfest to the meeting of 27 June 1931 of the Amsterdam Academy [7], and he also included them as Chapter IV of his dissertation, which he defended on 2 November 1931 at the University of Leiden [8].