For the case of , it is well-known from the discussion of angular momentum in any quantum mechanics textbook that irreducible representations can be labeled either by j, the highest weight (here, highest eigenvalue of ), or by , the eigenvalue of . The first of these requires making a choice (the z-axis) and looking at a specific vector in the representation, the second doesn’t. It was a physicist (Hendrik Casimir), who first recognized the existence of an analog of for general semi-simple Lie algebras, and the important role that this plays in representation theory.
The Casimir Operator
Recall that for a semi-simple Lie algebra one has a non-degenerate, invariant, symmetric bi-linear form , the Killing form, given by
If one starts with the Lie algebra of a compact group, this bilinear form is defined on , and negative-definite on . For a simple Lie algebra, taking the trace in a different representation gives the same bilinear form up to a constant. As an example, for the case , one can show that
here taking the trace in the fundamental representation as by complex matrices.
One can use the Killing form to define a distinguished quadratic element of , the Casimir element
where is an orthonormal basis with respect to the Killing form and is the dual basis. On any representation , this gives a Casimir operator
Note that, taking the representation to be the space of functions on the compact Lie group G, is an invariant second-order differential operator, (minus) the Laplacian.
is independent of the choice of basis, and belongs to , the subalgebra of invariant under the adjoint action. It turns out that , the center of . By Schur’s lemma, anything in the center must act on an irreducible representation by a scalar. One can compute the scalar for an irreducible representation as follows:
Choose a basis of with an orthonormal basis of the Cartan subalgebra , and elements of in the root-spaces of , orthonormal in the sense of satisfying
Then one has the following expression for :
To compute the scalar eigenvalue of this on an irreducible representation of highest weight , one can just act on a highest weight vector . On this vector the raising operators act trivially, and using the commutation relation
( is the element of satisfying ) one finds
where is half the sum of the positive roots, a quantity which keeps appearing in this story. Acting on one finds
Using the inner-product induced on by the Killing form, this eigenvalue can be written as:
In the special case , there is just one positive root, and one can take
Computing the Killing form, one finds
On a highest weight vector acts as
This is 1/2 times the physicist’s operator , and in the irreducible representation of spin , it acts with eigenvalue .
In the next posting in this series I’ll discuss the Harish-Chandra homomorphism, and the question of how the Casimir acts not just on , but on all of the cohomology . After that, taking note that the Casimir is in some sense a Laplacian, we’ll follow Dirac and introduce Clifford algebras and spinors in order to take its square root.