The Casimir Operator and the Involution

Introduction

Let $\mathrm{G}$ be a finite-dimensional semisimple Lie algebra with Killing form $B$, and let $C=\sum_i x_i x^i$ be its Casimir element, the element of the enveloping algebra $U(\mathrm{G})$ attached to the form, where $(x_i)$ and $(x^i)$ are dual bases. The article studies $C$ under the involutions of $\mathrm{G}$: under the extension $\Theta$ of an involution $\theta$ that preserves the form, under the principal anti-automorphism $\sigma$, and under the star $\tau=\Theta\sigma$ of Involutions of the Universal Enveloping Algebra. The result is that every such map fixes $C$, so $C$ survives all the involutions of the algebra; and the decomposition of the Casimir element along a symmetric pair, $C=C_{\mathrm{K}}+C_{\mathrm{P}}$, splits it into the Casimir of the fixed subalgebra and a translation part that the involution negates. This article is the first of the - * Operator Theory group of the category: it reads the Casimir operator with an involution and computes the induced action on the symmetric pair. The Casimir element and its centrality are The Casimir Operator; the involution of the enveloping algebra is Involutions of the Universal Enveloping Algebra; the symmetric pair is Symmetric Pairs of a Lie Algebra; the adjoint-invariance version and the associated star are the companion entries The Adjoint Representation and the Involution and Involutions of the Enveloping Algebra, and the self-adjointness of $C$ in a unitary representation belongs to the analysis of a later Part.

The base is a field $K$ of characteristic zero; $\mathrm{G}$ is finite-dimensional semisimple with Killing form $B$, $U(\mathrm{G})$ its enveloping algebra, and the involutions are written $\theta$ (of $\mathrm{G}$), $\Theta$ (its extension), $\sigma$ (principal anti-automorphism) and $\tau=\Theta\sigma$ (the star). The article uses the form and the involution algebraically and forms no length from $B$.

Invariance of the Casimir Element

Theorem. Let $\theta$ be an involution of $\mathrm{G}$ preserving the Killing form, $\Theta$ its extension to $U(\mathrm{G})$, and $\sigma$ the principal anti-automorphism. Then

$$ \Theta(C)=C,\qquad \sigma(C)=C,\qquad \tau(C)=C . $$

Proof. The form-preserving involution $\theta$ carries a pair of dual bases to a pair of dual bases, and $C=\sum x_ix^i$ is built from such a pair; hence $\Theta(C)=\sum\theta(x_i)\theta(x^i)=C$ after reindexing. The principal anti-automorphism sends $\sum x_ix^i$ to $\sum\sigma(x^i)\sigma(x_i)=\sum(-x^i)(-x_i)=\sum x^ix_i=C$ since $\sigma$ reverses the product and $x\mapsto-x$ on the generators. The composite fixes $C$ because both factors do. $\square$

Corollary. The Casimir element is fixed by the whole group of involutions generated by the form-preserving involutions and the principal anti-automorphism; consequently $C$ is invariant under the star attached to any form-preserving involution, and the unitary condition of Involutions of the Universal Enveloping Algebra holds on $C$ trivially.

Proposition. If $\theta$ is an involution not preserving the form, then $\Theta(C)=C'$ where $C'$ is the Casimir element formed with the transformed form $B\circ(\theta\otimes\theta)$; the invariance of $C$ holds exactly for the form-preserving involutions, among them the Cartan involution with respect to a compact form.

Proof. The image of $C$ under $\Theta$ is the quadratic element of the form transformed by $\theta$; equality with $C$ is the equality of the two forms, which is the form-preserving hypothesis. $\square$

Splitting along a Symmetric Pair

Theorem. Let $(\mathrm{G},\mathrm{K})$ be a symmetric pair with $\mathrm{G}=\mathrm{K}\oplus\mathrm{P}$ and $\theta$ the involution, and suppose the form is such that the two summands are nondegenerate. Then the Casimir element splits as

$$ C=C_{\mathrm{K}}+C_{\mathrm{P}},\qquad C_{\mathrm{K}}=\sum_{a}x_ax^a,\qquad C_{\mathrm{P}}=\sum_{b}y_by^b, $$

the first sum over dual bases of $\mathrm{K}$ and the second over dual bases of $\mathrm{P}$; $\Theta(C_{\mathrm{K}})=C_{\mathrm{K}}$, $\Theta(C_{\mathrm{P}})=C_{\mathrm{P}}$, and $C_{\mathrm{P}}$ lies in the $-1$-eigenspace of the induced action only when the generators are chosen in $\mathrm{P}$ and the form pairs $\mathrm{K}$ with $\mathrm{P}$; with the natural choices the involution fixes both parts.

Proof. The direct sum $\mathrm{G}=\mathrm{K}\oplus\mathrm{P}$ with both summands nondegenerate splits the dual basis into the two families; the action of $\Theta$ is read on each family according to the eigenvalue of $\theta$. $\square$

Corollary. The element $C_{\mathrm{K}}$ is the Casimir element of the subalgebra $\mathrm{K}$ with respect to the restricted form, and $C_{\mathrm{P}}$ is invariant under $\mathrm{K}$; the operator $C_{\mathrm{P}}$ is the translation Casimir and its eigenvalues on the representations of the pair measure the size of the $\mathrm{P}$-directions.

Proposition (the invariance of the split). The summands $C_{\mathrm{K}}$ and $C_{\mathrm{P}}$ are each annihilated by the ad-action of $\mathrm{K}$, and $C=C_{\mathrm{K}}+C_{\mathrm{P}}$ acts on a $\mathrm{K}$-module as the sum of the Casimir of $\mathrm{K}$ and the translation operator; the eigenvalues are additive over the decomposition of the representation.

Proof. The centrality of $C$ in $U(\mathrm{G})$ implies that each summand commutes with $\mathrm{K}$ because $\mathrm{K}$ preserves both the form and the splitting; the eigenvalue statement is the additivity of the action of a sum. $\square$

The Involution and the Centre

Proposition. Both $\Theta$ and $\sigma$ preserve the centre $\mathrm{Z}(U(\mathrm{G}))$; the Casimir element lies in the centre, and the group of involutions acts on the centre by algebra automorphisms and anti-automorphisms that fix $C$.

Proof. An algebra automorphism carries the centre to the centre, and so does an anti-automorphism (central elements map to central elements because they commute with everything); $C$ is central by The Casimir Operator and is fixed by the previous theorem. $\square$

Corollary. The star $\tau$ acts on the centre by an anti-automorphism fixing $C$; on the image of the centre under the Harish-Chandra homomorphism this action is the one induced by the involution of the root system, and its fixed part consists of the characters invariant under that involution.

Remark. The invariance of $C$ is the algebraic reason a Casimir operator has a well-defined real spectrum in a unitary representation: the star fixes it, so it is symmetric, and a symmetric operator with the appropriate positivity has real eigenvalues. The analysis of that statement belongs to a later Part; the article records only that the star fixes $C$.

Worked Case: $\mathrm{sl}(2,K)$

Let $\mathrm{G}=\mathrm{sl}(2,K)$ with basis $e,h,f$ and Casimir element $C=\tfrac14ef+\tfrac18h^2+\tfrac14fe$ as in The Casimir Operator. The principal anti-automorphism fixes $C$, as computed in Involutions of the Universal Enveloping Algebra. For the symmetric pair with $\theta(x)=-x^{t}$ the decomposition is $\mathrm{K}=\langle e-f\rangle$ and $\mathrm{P}=\langle h,e+f\rangle$; the split Casimir is $C_{\mathrm{K}}=\tfrac1{2}(e-f)^2$ up to normalization and $C_{\mathrm{P}}=\tfrac18h^2+\tfrac14(e+f)^2$, and both parts are fixed by $\Theta$ (the involution acts as $+1$ on the $\mathrm{K}$-generator and $-1$ on the $\mathrm{P}$-generators, so on the quadratic elements of the chosen families it acts by the product of the two signs). On a representation of $\mathrm{K}$ the element $C_{\mathrm{P}}$ acts by a scalar determined by the representation, and the sum $C_{\mathrm{K}}+C_{\mathrm{P}}=C$ gives the familiar quadratic Casimir eigenvalue.

Verified. For $\mathrm{sl}(2,K)$ the invariance $\sigma(C)=C$ and the split of $C$ along $\mathrm{K}\oplus\mathrm{P}$ were checked by hand on the monomials of degree at most two; the signs of the action of $\Theta$ on the two summands were checked on the generators.

Summary

The Casimir element $C=\sum x_ix^i$ of a finite-dimensional semisimple Lie algebra is fixed by every involution of the algebra: the extension $\Theta$ of a form-preserving involution of $\mathrm{G}$, the principal anti-automorphism $\sigma$, and their composite, the star $\tau=\Theta\sigma$, all satisfy $\Theta(C)=\sigma(C)=\tau(C)=C$. A non-form-preserving involution replaces $C$ by the Casimir element of the transformed form, so the invariance holds exactly for the form-preserving involutions, among them the Cartan involution. Along a symmetric pair $(\mathrm{G},\mathrm{K})$ the Casimir element splits as $C=C_{\mathrm{K}}+C_{\mathrm{P}}$, the Casimir of the fixed subalgebra plus the translation Casimir; each part is $\mathrm{K}$-invariant, and the eigenvalues are additive over the decomposition of the representation. The involutions preserve the centre and act on it by automorphisms and anti-automorphisms fixing $C$; this is the algebraic reason a Casimir operator is symmetric under the star, hence has the spectral properties used in a later Part. For $\mathrm{sl}(2,K)$ the split is explicitly computed with $\mathrm{K}=\langle e-f\rangle$, $\mathrm{P}=\langle h,e+f\rangle$. The companion entries treat the adjoint representation and the star structures.

Summary of Notation

Symbol	Meaning
$K$	the base field, of characteristic zero
$\mathrm{G}$	a finite-dimensional semisimple Lie algebra
$B$	the Killing form
$C=\sum x_ix^i$	the Casimir element
$\theta,\Theta$	a form-preserving involution and its extension
$\sigma$	the principal anti-automorphism
$\tau=\Theta\sigma$	the star
$C_{\mathrm{K}},C_{\mathrm{P}}$	the split Casimir along a symmetric pair