For the case of [tex]G=SU(2)[\/tex], 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 [tex]J_3[\/tex] ), or by [tex]j(j+1)[\/tex], the eigenvalue of [tex]\\mathbf{J\\cdot J}[\/tex]. 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 [tex]\\mathbf{J\\cdot J}[\/tex] for general semi-simple Lie algebras, and the important role that this plays in representation theory.<\/p>\n
The Casimir Operator<\/strong><\/p>\n Recall that for a semi-simple Lie algebra [tex]\\mathfrak g[\/tex] one has a non-degenerate, invariant, symmetric bi-linear form [tex](\\cdot,\\cdot)[\/tex], the Killing form, given by<\/p>\n [tex](X,Y)= tr(ad(X)ad(Y))[\/tex] <\/p>\n If one starts with [tex]\\mathfrak g[\/tex] the Lie algebra of a compact group, this bilinear form is defined on [tex]\\mathfrak g_{\\mathbf C}[\/tex], and negative-definite on [tex]\\mathfrak g[\/tex]. 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 [tex]\\mathfrak g_{\\mathbf C}={\\mathfrak{sl}(n,\\mathbf C)}[\/tex], one can show that <\/p>\n ([tex]X,Y)=2n\\ tr(XY)[\/tex]<\/p>\n here taking the trace in the fundamental representation as [tex]n[\/tex] by [tex]n[\/tex] complex matrices. [tex]\\Omega=\\sum_iX_iX^i[\/tex]<\/p>\n where [tex]X_i[\/tex] is an orthonormal basis with respect to the Killing form and [tex]X^i[\/tex] is the dual basis. On any representation [tex]V[\/tex], this gives a Casimir operator<\/p>\n [tex]\\Omega_V=\\sum_i\\pi(X_i)\\pi(X^i)[\/tex]<\/p>\n Note that, taking the representation [tex]V[\/tex] to be the space of functions [tex]C^\\infty(G)[\/tex] on the compact Lie group G, [tex]\\Omega_V[\/tex] is an invariant second-order differential operator, (minus) the Laplacian.<\/p>\n [tex]\\Omega[\/tex] is independent of the choice of basis, and belongs to [tex]U(\\mathfrak g)^{\\mathfrak g}[\/tex], the subalgebra of [tex]U(\\mathfrak g)[\/tex] invariant under the adjoint action. It turns out that [tex]U(\\mathfrak g)^{\\mathfrak g}=Z(\\mathfrak g)[\/tex], the center of [tex]U(\\mathfrak g)[\/tex]. By Schur’s lemma, anything in the center [tex]Z(\\mathfrak g)[\/tex] must act on an irreducible representation by a scalar. One can compute the scalar for an irreducible representation [tex](\\pi,V)[\/tex] as follows:<\/p>\n Choose a basis [tex](H_i, X_{\\alpha},X_{-\\alpha})[\/tex] of [tex]\\mathfrak g_{\\mathbf C}[\/tex] with [tex]H_i[\/tex] an orthonormal basis of the Cartan subalgebra [tex]\\mathfrak t_{\\mathbf C}[\/tex], and [tex]X_{\\pm\\alpha}[\/tex] elements of [tex]\\mathfrak n^{\\pm}[\/tex] in the [tex]\\pm\\alpha[\/tex] root-spaces of [tex]\\mathfrak g_{\\mathbf C}[\/tex], orthonormal in the sense of satisfying <\/p>\n [tex](X_{\\alpha},X_{-\\alpha})=1[\/tex]<\/p>\n Then one has the following expression for [tex]\\Omega[\/tex]:<\/p>\n [tex]\\Omega=\\sum_i H_i^2 + \\sum_{+\\ roots} (X_{\\alpha} X_{-\\alpha} +X_{-\\alpha}X_{\\alpha})[\/tex]<\/p>\n To compute the scalar eigenvalue of this on an irreducible representation [tex](\\pi,V_{\\lambda})[\/tex] of highest weight [tex]\\lambda[\/tex], one can just act on a highest weight vector [tex]v\\in V^{\\lambda}=V^{\\mathfrak n^+}[\/tex]. On this vector the raising operators [tex]\\pi(X_{\\alpha})[\/tex] act trivially, and using the commutation relation<\/p>\n [tex][X_{\\alpha},X_{-\\alpha}]=H_{\\alpha}[\/tex]<\/p>\n ([tex]H_{\\alpha}[\/tex] is the element of [tex]\\mathfrak t_{\\mathbf C} [\/tex] satisfying [tex](H,H_{\\alpha})=\\alpha(H)[\/tex]) one finds<\/p>\n [tex]\\Omega=\\sum_i H_i^2 + \\sum_{+\\ roots}H_{\\alpha}= \\sum_i H_i^2 +2H_{\\rho}[\/tex]<\/p>\n where [tex]\\rho[\/tex] is half the sum of the positive roots, a quantity which keeps appearing in this story. Acting on [tex]v\\in V^{\\lambda}[\/tex] one finds<\/p>\n [tex]\\Omega_{V_{\\lambda}}v=(\\sum_i\\lambda(H_i)^2+2\\lambda (H_{\\rho}))v[\/tex]<\/p>\n Using the inner-product [tex]< \\cdot,\\cdot>[\/tex] induced on [tex]\\mathfrak t^*[\/tex] by the Killing form, this eigenvalue can be written as:<\/p>\n [tex]< \\lambda,\\lambda>+2< \\lambda,\\rho>=||\\lambda+\\rho||^2- ||\\rho||^2[\/tex]<\/p>\n In the special case [tex]\\mathfrak g = \\mathfrak {su}(2),\\ \\mathfrak g_{\\mathbf C}=\\mathfrak sl(2,\\mathbf C)[\/tex], there is just one positive root, and one can take<\/p>\n [tex]H_1=h=\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix},\\ X_{\\alpha}=e=\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix},\\ X_{-\\alpha}=f=\\begin{pmatrix}0&0\\\\1&0\\end{pmatrix}[\/tex]<\/p>\n Computing the Killing form, one finds<\/p>\n [tex](h,h)=8,\\ (e,f)=4[\/tex]<\/p>\n and<\/p>\n [tex]\\Omega=\\frac{1}{8}h^2 + \\frac{1}{4}(ef +fe)=\\frac{1}{8}h^2 + \\frac{1}{4}(h +2fe)[\/tex]<\/p>\n On a highest weight vector [tex]\\Omega[\/tex] acts as<\/p>\n [tex]\\Omega=\\frac{1}{8}h^2 + \\frac{1}{4}h=\\frac{1}{8}h(h+2)=\\frac{1}{2}(\\frac{h}{2}(\\frac{h}{2} +1))[\/tex]<\/p>\n This is 1\/2 times the physicist’s operator [tex]\\mathbf{J\\cdot J}[\/tex], and in the irreducible representation [tex]V_n[\/tex] of spin [tex] j=n\/2[\/tex], it acts with eigenvalue [tex]\\frac{1}{2}j(j+1)[\/tex].<\/p>\n 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 [tex]V^{\\mathfrak n^+}=H^0(\\mathfrak n^+,V)[\/tex], but on all of the cohomology [tex]H^*(\\mathfrak n^+,V)[\/tex]. 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.<\/p>\n","protected":false},"excerpt":{"rendered":" For the case of [tex]G=SU(2)[\/tex], 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 [tex]J_3[\/tex] ), or by [tex]j(j+1)[\/tex], … Continue reading
\nOne can use the Killing form to define a distinguished quadratic element [tex]\\Omega[\/tex] of [tex]U(\\mathfrak g)[\/tex], the Casimir element<\/p>\n