The Casimir Operator of a Lie Group

Introduction

On a semisimple Lie algebra the invariant bilinear form produces, from any basis and its dual basis, a single element of the universal enveloping algebra, the Casimir element $\Omega = \sum_i X_iX^i$. It is central, so it defines a bi-invariant differential operator on the group, and in every irreducible representation it acts by a scalar; the scalar is the quadratic invariant of the representation, the first of the labels by which the irreducible representations are indexed. The operator is the Laplacian of a bi-invariant metric when the group is compact, and it is the model of every bi-invariant operator, since on a semisimple algebra the centre of the enveloping algebra is generated by the Casimir elements of the simple summands.

This article treats the Casimir element and the operator it defines. It is the second article of the - Operator Theory group of the category; the invariant differential operators and the identification of the enveloping algebra with them are from Operators on a Lie Group, the centre of the enveloping algebra and the first example of the Casimir are stated there, and the eigenvalue theory is completed by The Involution on the Enveloping Algebra of a Lie Group and Unitary Representations of a Lie Group of the - * Theory group.

The article assumes the Lie algebra $\mathrm{G}$ of a Lie group and the adjoint representation from Lie Algebras, the invariant bilinear forms, the Killing form and the trace form from Structure of Lie Algebras, the universal enveloping algebra with the Poincaré--Birkhoff--Witt theorem from Universal Enveloping Algebras, and the representation theory of a semisimple algebra, with Schur's lemma and the highest weight classification, from Representations of Lie Algebras. No topology is used beyond the smooth structure of Operators on a Lie Group, and no metric is fixed until the compact case is discussed at the end.

The Casimir Element

The Invariant Form

Let $\mathrm{G}$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero, and let $B$ be a symmetric bilinear form on $\mathrm{G}$ that is invariant,

$$ B([X,Y], Z) + B(Y, [X,Z]) = 0 \qquad (X,Y,Z \in \mathrm{G}), $$

equivalently $B(\operatorname{ad}_X Y, Z) + B(Y, \operatorname{ad}_X Z) = 0$. The Killing form $B(X,Y) = \operatorname{tr}(\operatorname{ad}_X\operatorname{ad}_Y)$ is invariant, and every invariant symmetric form on a simple algebra is a scalar multiple of it; on a semisimple algebra the invariant symmetric forms are exactly the forms that are diagonal with respect to the decomposition into simple summands.

Definition. Let $B$ be an invariant symmetric bilinear form on $\mathrm{G}$, and assume it nondegenerate. For a basis $(X_i)$ of $\mathrm{G}$ the dual basis $(X^i)$ is defined by $B(X_i, X^j) = \delta_i^j$, where $\delta_i^j$ is the Kronecker delta. The Casimir element of the pair $(\mathrm{G}, B)$ is

$$ \Omega_B = \sum_i X_i X^i \ \in\ U(\mathrm{G}) . $$

Proposition (independence of the basis). The element $\Omega_B$ does not depend on the choice of the basis $(X_i)$.

Proof. A second basis is $X'_j = \sum_i c_j{}^i X_i$ with $c$ invertible, and its dual basis is $X'{}^j = \sum_i d_i{}^j X^i$ with $d = (c^{-1})^{\mathsf{T}}$; substituting gives $\sum_j X'_jX'{}^j = \sum_{j,i,l} c_j{}^i d_l{}^j X_iX^l = \sum_{i,l}\bigl(\sum_j c_j{}^i d_l{}^j\bigr)X_iX^l = \sum_i X_iX^i$, since $\sum_j c_j{}^i d_l{}^j = \delta_l^i$.

The Dependence on the Form

Proposition (rescaling the form rescales the element). If $B' = \lambda B$ with $\lambda \in k^\times$, then $X'{}^i = \lambda^{-1}X^i$ and $\Omega_{B'} = \lambda^{-1}\Omega_B$. Hence the Casimir element is determined by the form only up to a scalar, and a normalisation of the form is part of the data. On a simple algebra, where the invariant forms are the scalar multiples of the Killing form, the Casimir element is determined up to a scalar, and the conventional normalisations are the one by the Killing form and the one by the trace form $\operatorname{tr}(\rho(X)\rho(Y))$ of a faithful representation $\rho$.

Proof. $B'(X_i, \lambda^{-1}X^j) = \lambda B(X_i, \lambda^{-1}X^j) = \delta_i^j$, so the dual basis of $B'$ is $\lambda^{-1}X^j$; the sum scales accordingly. The classification of the invariant forms on a simple algebra is that of Structure of Lie Algebras, and the trace form of a faithful representation is a nonzero multiple of the Killing form.

Centrality

Theorem (the Casimir element is central). For every $Z \in \mathrm{G}$ the derivation $\operatorname{ad}_Z$ annihilates $\Omega_B$, and hence

$$ \Omega_B \in Z(U(\mathrm{G})) . $$

Proof. In the enveloping algebra the derivation $\operatorname{ad}_Z = [Z, \cdot]$ acts by the Leibniz rule, so

$$ \operatorname{ad}_Z\Omega_B = \sum_i \bigl([Z,X_i]X^i + X_i[Z,X^i]\bigr) . $$

Writing $[Z,X_i] = \sum_j c_{Zi}{}^j X_j$ and using the invariance of $B$ in the form $B([Z,X_i], X^j) = -B(X_i, [Z,X^j])$, one gets $[Z,X^i] = -\sum_j c_{Zj}{}^i X^j$; substituting and renaming the indices, the two sums cancel. Since the $\operatorname{ad}_Z$ generate the adjoint action and the centre of $U(\mathrm{G})$ is the common kernel of the derivations $\operatorname{ad}_Z$, the element is central.

Corollary (the Casimir operator). Under the isomorphism $U(\mathrm{G}) \to D_L(G)$ of Operators on a Lie Group, the central element $\Omega_B$ corresponds to a bi-invariant differential operator, the Casimir operator, written $\Omega_B f = \sum_i \tilde X_i\tilde X^i f$ on functions and $u \mapsto \Omega_B u$ on the enveloping algebra.

The Casimir of a Representation

The Quadratic Invariant

Let $\rho : \mathrm{G} \to \operatorname{End}_k(V)$ be a finite-dimensional representation, and write $\rho(\Omega_B) = \sum_i \rho(X_i)\rho(X^i)$.

Theorem (the Casimir of a representation is an intertwining scalar). $\rho(\Omega_B)$ commutes with $\rho(\mathrm{G})$; it is central in the image algebra, and if $\rho$ is irreducible then $\rho(\Omega_B) = c_\rho\,\mathrm{id}_V$ for a scalar $c_\rho \in k$.

Proof. The centrality of $\Omega_B$ gives $\rho(\Omega_B)\rho(X) = \rho(\Omega_BX) = \rho(X\Omega_B) = \rho(X)\rho(\Omega_B)$; for an irreducible representation a central operator is a scalar by Schur's lemma, which is the form of Representations of Lie Algebras.

Theorem (the trace of the Casimir). Let $\rho$ be a finite-dimensional representation and let $B_\rho(X,Y) = \operatorname{tr}(\rho(X)\rho(Y))$ be its trace form. With the Casimir element formed for the form $B_\rho$, one has

$$ \operatorname{tr}\bigl(\rho(\Omega_{B_\rho})\bigr) = \dim_k \mathrm{G} . $$

In particular, for the trace form of a faithful representation and for the Killing form of the adjoint representation, the trace of the Casimir is the dimension of $\mathrm{G}$.

Proof. $\operatorname{tr}(\rho(\Omega_{B_\rho})) = \sum_i \operatorname{tr}(\rho(X_i)\rho(X^i)) = \sum_i B_\rho(X_i, X^i) = \sum_i \delta_i^i = \dim_k\mathrm{G}$.

The Eigenvalue on a Highest Weight Module

Let $\mathrm{G}$ be a split semisimple algebra over an algebraically closed field of characteristic zero, let $\mathfrak{h}$ be a Cartan subalgebra and let $V_\lambda$ be the irreducible representation of highest weight $\lambda$. With the normalisation in which the Casimir is formed by the Killing form and $\langle\cdot,\cdot\rangle$ is the induced form on $\mathfrak{h}^*$, the quadratic invariant is

$$ c_\lambda = \langle \lambda, \lambda + 2\varrho\rangle , $$

where $\varrho$ is the Weyl vector, half the sum of the positive roots.

Theorem. $\rho_{V_\lambda}(\Omega) = c_\lambda\,\mathrm{id}$.

Proof. The Casimir element can be written as $\Omega = \sum_i h_ih^i + 2\sum_{\alpha>0}\frac{1}{\langle\alpha,\alpha\rangle}X_\alpha X_{-\alpha}$ in a root basis, with $(h_i)$ a basis of $\mathfrak{h}$ and $(h^i)$ the dual basis. On the highest weight vector $v_\lambda$ the elements $X_\alpha$ with $\alpha > 0$ annihilate and the $h_i$ act by $\lambda$; computing $\Omega v_\lambda$ and using $\sum_{\alpha>0}\alpha = 2\varrho$ gives the displayed scalar. The statement is the standard computation, and it is developed with the highest weight classification in Representations of Lie Algebras.

Corollary. The Casimir eigenvalue distinguishes the irreducible representations only up to the finite ambiguity of the level sets of $\lambda \mapsto \langle\lambda,\lambda+2\varrho\rangle$; the full label is the highest weight, of which $c_\lambda$ is the quadratic part, and the complete set of Casimir elements of a semisimple algebra are the generators of the centre of $U(\mathrm{G})$.

Proof. The eigenvalue is a quadratic function of $\lambda$ and its fibres are infinite; the algebra generated by the eigenvalues is the centre of the enveloping algebra by the Harish-Chandra isomorphism, which associates to each central element its eigenvalue on the highest weight modules.

The Casimir as a Laplacian

The Bi-Invariant Metric

Definition. Let $G$ be a connected Lie group whose algebra $\mathrm{G}$ is compact and semisimple, and let $B$ be an invariant symmetric form that is negative definite, such as the negative of the Killing form. The bi-invariant metric associated with $B$ is the Riemannian metric obtained by translating $B$ to every tangent space by the left translations; it is also right-invariant because $B$ is invariant under the adjoint action.

Theorem. For a compact semisimple group with the bi-invariant metric of $-B$, the Casimir operator $\Omega_{-B} = -\sum_i \tilde X_i\tilde X^i$ is the Laplace--Beltrami operator of the metric, up to the normalisation of the form.

Proof. In an orthonormal frame the Laplace--Beltrami operator of a bi-invariant metric is the sum of the squares of the orthonormal left-invariant fields, because the Levi-Civita connection of a bi-invariant metric is given by $2\nabla_XY = [X,Y]$ and the divergence of a left-invariant field vanishes at $e$; the sum is $\Omega_{-B}$ in the orthonormal basis. The precise identification of the Laplacian and its spectral consequences are the subject of the harmonic analysis of Harmonic Analysis on Groups, and the representation-theoretic diagonalisation is that of The Regular Representation of a Lie Group.

The Casimir of a Symmetric Pair

Definition. Let $\mathrm{G} = \mathrm{K}\oplus\mathrm{P}$ be the Cartan decomposition of a real semisimple algebra for an involution $\theta$, and let $B$ be an invariant form negative definite on $\mathrm{K}$ and positive definite on $\mathrm{P}$. The Casimir operator of the pair is the sum of the two partial Casimir elements, $\Omega = \Omega_{\mathrm{K}} + \Omega_{\mathrm{P}}$, of the restrictions of $B$ to the two summands.

Proposition. $\Omega_{\mathrm{K}}$ and $\Omega_{\mathrm{P}}$ are each invariant under $\mathrm{K}$, and $\Omega_{\mathrm{K}}$ is central in $U(\mathrm{K})$; the sum $\Omega$ is the Casimir element of $\mathrm{G}$, and its decomposition separates the operators that act on the functions constant on the orbits of $\mathrm{K}$ from the radial part.

Proof. The involution $\theta$ is an automorphism, so it preserves $B$; the summands are the $\pm1$-eigenspaces and therefore orthogonal, so the dual basis of $\mathrm{G}$ with respect to $B$ is the union of the dual bases of the two summands; the invariance of $B$ gives the invariance of each partial sum under its own algebra, and the central statements follow from the centrality of $\Omega$ and the decomposition.

The Centre of the Enveloping Algebra

The Casimir-Type Invariants

Definition. A Casimir-type invariant of order $k$ is a central element of $U(\mathrm{G})$ that is a sum of products of $k$ elements of $\mathrm{G}$ and of lower order; the Casimir element of an invariant form is the invariant of order two determined by the form, and its construction is the one of the first sections.

Proposition. For a semisimple algebra the centre $Z(U(\mathrm{G}))$ is a polynomial algebra in $\operatorname{rank}\mathrm{G}$ generators, and the generators are the Casimir-type invariants; the first generator is the Casimir element of the Killing form, and the invariants of higher order are obtained from the invariant polynomials on the algebra by symmetrisation.

Proof. The symmetric algebra $S(\mathrm{G})$ is the graded algebra of the polynomial functions on $\mathrm{G}^{*}$, its invariants form a polynomial algebra in $\operatorname{rank}\mathrm{G}$ generators by the Chevalley theorem, and the symmetrisation identifies the invariants of $S(\mathrm{G})$ with the centre of $U(\mathrm{G})$ after the shift of the Harish-Chandra homomorphism; the Casimir element is the image of the invariant quadratic form, which is the first generator.

The Harish-Chandra Isomorphism

Theorem. Let $\mathrm{H}$ be a Cartan subalgebra of the semisimple algebra $\mathrm{G}$, let $\Delta^{+}$ be a system of positive roots and let $\varrho = \tfrac12\sum_{\alpha\in\Delta^{+}}\alpha$. The Harish-Chandra homomorphism is the algebra homomorphism

$$ \gamma : Z(U(\mathrm{G}))\longrightarrow S(\mathrm{H}) , $$

defined by projecting a central element on the Cartan part of its Poincaré--Birkhoff--Witt decomposition and shifting the result by $\varrho$; it is an isomorphism of $Z(U(\mathrm{G}))$ onto the Weyl-invariant symmetric algebra $S(\mathrm{H})^{W}$. On the irreducible representation of highest weight $\lambda$ the central element $z$ acts by the scalar $\gamma(z)(\lambda+\varrho)$, and the Casimir element corresponds to the invariant $\lVert\lambda+\varrho\rVert^{2}-\lVert\varrho\rVert^{2}$.

Proof. The projection onto the Cartan part is well defined on the centre because the lower terms of a central element have a vanishing projection; the shift by $\varrho$ makes the projection an algebra homomorphism, the computation on the Verma module giving the character of the centre, and the image is the Weyl-invariant part because the centre acts by the same character on the Weyl-group orbit of a weight; the bijectivity is the Chevalley restriction theorem. The eigenvalue is computed from the Casimir element acting on the highest-weight vector.

Corollary. The central characters of the irreducible representations are the homomorphisms $z\mapsto\gamma(z)(\lambda+\varrho)$, the Weyl-group orbit of the highest weight gives the same central character, and the centre separates the irreducible representations by the infinitesimal character $\lambda+\varrho$.

Proof. The central character is the evaluation of $\gamma(z)$ at $\lambda+\varrho$ by the theorem, and it depends only on the Weyl-group orbit of $\lambda+\varrho$; the separation is the bijectivity of $\gamma$ and the identification of the characters with the points of the orbit space.

Summary

For a finite-dimensional Lie algebra with an invariant nondegenerate symmetric form $B$, the choice of a basis $(X_i)$ with dual basis $(X^i)$ determines the Casimir element $\Omega_B = \sum_i X_iX^i \in U(\mathrm{G})$, independent of the basis and depending on the form only by the inverse scalar of a rescaling. It is central, $\operatorname{ad}_Z\Omega_B = 0$, so it defines a bi-invariant differential operator, the Casimir operator, on the group; in a representation it commutes with the image of the algebra and acts by a scalar on every irreducible representation, and with the trace form of a finite-dimensional representation its trace is $\dim_k\mathrm{G}$. On the irreducible representation of highest weight $\lambda$ the scalar is the quadratic invariant $\langle\lambda,\lambda+2\varrho\rangle$, and on a compact semisimple group with a bi-invariant metric the Casimir operator is the Laplace--Beltrami operator of the metric. The central elements of the enveloping algebra are the Casimir-type invariants, and their eigenvalues on the irreducible representations generate the centre by the Harish-Chandra isomorphism.

Summary of Notation

Symbol	Meaning
$B$	an invariant nondegenerate symmetric bilinear form on $\mathrm{G}$
$(X_i)$, $(X^i)$	a basis and its dual basis, $B(X_i, X^j) = \delta_i^j$
$\Omega_B = \sum_i X_iX^i$	the Casimir element
$\Omega_B \in Z(U(\mathrm{G}))$	centrality
$\rho(\Omega_B)$	the Casimir of a representation, scalar when irreducible
$B_\rho(X,Y) = \operatorname{tr}(\rho(X)\rho(Y))$	the trace form of a representation
$\operatorname{tr}\rho(\Omega_{B_\rho}) = \dim_k\mathrm{G}$	the trace of the Casimir
$c_\lambda = \langle\lambda,\lambda+2\varrho\rangle$	the quadratic invariant of the highest weight $\lambda$
$\varrho$	the Weyl vector, half the sum of the positive roots
$\Omega_{-B}$	the Laplacian of a compact group for a bi-invariant metric
$\Omega_{\mathrm{K}} + \Omega_{\mathrm{P}}$	the decomposition of the Casimir for a Cartan pair