The Regular Representation of a Lie Group

Introduction

A Lie group acts on the space of functions on itself in two ways, by the left and by the right translations, and the two actions commute; the pair of them is the regular representation of the group, and its infinitesimal form is the representation of the Lie algebra by the left- and right-invariant vector fields, which extends to the universal enveloping algebra. The left regular representation is the one whose derived representation is injective on the algebra, and the bi-invariant operators — the elements of the centre of the enveloping algebra — are exactly the operators that commute with both actions. The decomposition of the regular representation is the harmonic analysis of the group, and the two-sided action is the model of every operator built from the product.

This article treats the regular representation of a Lie group: the two translation actions and their commutation, the derived representation of the Lie algebra and its extension to the enveloping algebra, the two-sided action and the group algebra, and the compact case with the Peter--Weyl decomposition. It is the fourth article of the - Operator Theory group of the category; the invariant differential operators and their identification with the enveloping algebra are from Operators on a Lie Group, the exponential map is The Exponential Map as an Operator, and the unitary structure and the decomposition are Unitary Representations of a Lie Group and Harmonic Analysis on Groups, the latter written and cited for the analysis.

The article assumes the Lie group and its smooth structure from Lie Groups, the left- and right-invariant vector fields and the exponential from The Lie Algebra and the Exponential Map, the universal enveloping algebra from Universal Enveloping Algebras, and the Haar integral from Locally Compact Groups and Haar Measure. It uses the group algebra $k[G]$ of a topological group only by name, for the comparison with the enveloping algebra; the $L^2$ theory, the irreducible decomposition and the Plancherel measure belong to Unitary Representations of a Lie Group and to Harmonic Analysis on Groups, and are named here and not developed.

The Two Translation Actions

The Actions on Functions

Definition. The left regular representation and the right regular representation of $G$ are the actions on the algebra $C^\infty(G)$ of smooth functions given by

$$ (L_gf)(x) = f(g^{-1}x), \qquad (R_gf)(x) = f(xg) \qquad (g,x\in G) . $$

The two-sided action is the homomorphism $G\times G \to \operatorname{Aut}(C^\infty(G))$, $(g,h)\mapsto L_gR_h$.

Proposition. Each of the two actions is an action of $G$ by algebra automorphisms of $C^\infty(G)$, the two actions commute, and the two-sided action has kernel the anti-diagonal copy of the centre of $G$.

Proof. Pullback along a diffeomorphism is an algebra automorphism, and $L_{gh} = L_gL_h$, $R_{gh} = R_gR_h$; the commutation is $(L_gR_hf)(x) = f(g^{-1}xh) = (R_hL_gf)(x)$. The kernel is the set of $(g,h)$ with $L_gR_h = \mathrm{id}$, which acts on functions by $f(g^{-1}xh) = f(x)$ for all $f$, hence $g^{-1}xh = x$ for all $x$, so $h = g^{-1}$ and $g$ is central; this is the anti-diagonal copy of the centre.

The Derived Representation

Definition. The derived representation of the Lie algebra is the linear map

$$ \mathrm{G} \longrightarrow \operatorname{End}_\mathbb{C}(C^\infty(G)), \qquad X \longmapsto \tilde X , $$

sending $X$ to the left-invariant vector field $\tilde X$ of Operators on a Lie Group; it is a representation of the Lie algebra by derivations of $C^\infty(G)$, $[\tilde X, \tilde Y] = \widetilde{[X,Y]}$.

Theorem (the derived representation is the derivative of the regular representation). For $X \in \mathrm{G}$ and $f \in C^\infty(G)$,

$$ \tilde X f = \frac{d}{dt}\Big|_{t=0} L_{\exp(tX)}f . $$

Hence the derived representation of the left regular representation is the representation by the left-invariant fields, and the right regular representation has the derived representation $X \mapsto -\tilde X^{R}$, where $\tilde X^{R}$ is the right-invariant field equal to $X$ at the identity.

Proof. The value is $\frac{d}{dt}\big|_{t=0} f(\exp(-tX)x)$, which is the definition of the left-invariant field. The right statement is the same computation with $R_{\exp(tX)}$ and the right-trivialisation, the sign being that of the inverse in the flow.

The Extension to the Enveloping Algebra

Theorem. The derived representation extends to an algebra homomorphism

$$ U(\mathrm{G}) \longrightarrow \operatorname{End}_\mathbb{C}(C^\infty(G)), \qquad X_1 \cdots X_k \longmapsto \tilde X_1 \cdots \tilde X_k , $$

whose image is the algebra $D_L(G)$ of left-invariant differential operators; it is injective, and its image on the right-invariant side is the opposite algebra $D_R(G)$.

Proof. This is the universal property of the enveloping algebra applied to the Lie algebra homomorphism $X \mapsto \tilde X$, together with the isomorphism $U(\mathrm{G}) \cong D_L(G)$ of Operators on a Lie Group. The right-invariant side is the same statement with the opposite multiplication, since the right-invariant fields compose in the reverse order.

Corollary (the derived representations commute). For all $u, v \in U(\mathrm{G})$ and the corresponding left- and right-invariant operators $\tilde u \in D_L(G)$, $\tilde v \in D_R(G)$, one has $\tilde u\tilde v = \tilde v\tilde u$.

Proof. The left and right translations of the group commute and generate the two actions, so their differential operators commute; equivalently, the two-sided action differentiates to two commuting representations of the algebra and of its opposite.

The Two-Sided Action and the Group Algebra

The Two-Sided Action on Functions

Definition. The two-sided action of $U(\mathrm{G})\otimes U(\mathrm{G})$ on $C^\infty(G)$ is the extension of $(X,Y)\mapsto \tilde X - \tilde Y^{R}$, sending $u\otimes v$ to the operator $\tilde u\,\tilde v^{R}$; it is an algebra homomorphism from $U(\mathrm{G})\otimes U(\mathrm{G})$ to $\operatorname{End}_\mathbb{C}(C^\infty(G))$.

Theorem (the bi-invariant operators are the invariants). An operator $D \in \operatorname{End}_\mathbb{C}(C^\infty(G))$ commutes with every left and every right translation if and only if it lies in the image of the centre $Z(U(\mathrm{G}))$,

$$ D_L(G)\cap D_R(G) = \tilde Z(U(\mathrm{G})) . $$

Proof. This is the identification of the bi-invariant differential operators with the centre of the enveloping algebra from Operators on a Lie Group, restated as the statement that the operators fixed by the two-sided action are the elements of the centre. An operator commuting with all translations is in particular left-invariant and right-invariant, hence bi-invariant; the converse is the centrality.

The Comparison with the Group Algebra

Proposition. The group algebra $k[G]$ acts on functions by convolution, $f\mapsto f*h$ for $h\in k[G]$, and this action commutes with the left translations; the enveloping algebra acts on functions by the left-invariant differential operators $\tilde u$, and the two actions are related by the fact that the distribution supported at $e$ corresponding to $u$ is the limit of the group algebra elements approaching the identity. The enveloping algebra is not a subalgebra of the group algebra: it is the algebra of the distributions supported at the identity, by Operators on a Lie Group.

Proof. The convolution action and its commutation with the left translations are the convolution theorem of Operators on a Lie Group; the distribution realisation of the enveloping algebra is the same theorem, read for the distributions supported at the identity rather than for the smooth functions.

The Compact Case

The Peter--Weyl Decomposition

Theorem (Peter--Weyl). Let $G$ be compact. The left regular representation on $L^2(G)$ decomposes as the Hilbert direct sum

$$ L^2(G) = \bigoplus_{\pi\in\widehat{G}} V_\pi\otimes V_\pi^{*}, $$

over the set $\widehat{G}$ of equivalence classes of irreducible unitary representations, each occurring with multiplicity its dimension; the matrix coefficients $\pi\mapsto \langle \pi(g)v,w\rangle$ span a dense subspace. The decomposition is stated and proved in Unitary Representations of a Lie Group and in Harmonic Analysis on Groups; the operator-theoretic content is that the regular representation is the direct integral of the irreducible unitary representations.

Proof (statement). The proof is by the spectral theorem for the compact self-adjoint convolution operators, which is the analysis of Harmonic Analysis on Groups; the operator layer uses only the decomposition, and the Casimir operator of The Casimir Operator of a Lie Group acts on each summand by the scalar attached to $\pi$.

Theorem (the Casimir on the regular representation). On a compact semisimple group the Casimir operator of The Casimir Operator of a Lie Group is the Laplacian of a bi-invariant metric, its eigenspaces are the irreducible summands of the Peter--Weyl decomposition, and its eigenvalue on the summand of highest weight $\lambda$ is the quadratic invariant $\langle\lambda,\lambda+2\varrho\rangle$.

Proof. The Casimir is central, hence bi-invariant, hence acts on each irreducible summand by a scalar by Schur's lemma; the scalar is computed in The Casimir Operator of a Lie Group, and the identification with the Laplacian of a bi-invariant metric is the same article. The analytic consequences — the heat kernel, the Weyl character formula and the asymptotics — are the subject of Harmonic Analysis on Groups.

Examples

The Additive Group

Let $G = \mathbb{R}^n$. The left and right translations coincide with the translations of the vector space, $L_gf(x) = f(x-g) = R_{-g}f(x)$, the derived representation is $X\mapsto X\cdot\nabla$, and the enveloping algebra is the algebra of constant coefficient differential operators; the Fourier transform diagonalises the regular representation, and the decomposition is the direct integral over $\mathbb{R}^n$ of the characters, the Plancherel theorem of Harmonic Analysis on Groups.

A Compact Group

Let $G = SU(2)$. The irreducible unitary representations are the symmetric powers of the defining representation, the regular representation is the Hilbert sum of the $\pi_n$ with multiplicity $n+1$, and the Casimir operator acts by the scalar $-\frac14 n(n+2)$ in a standard normalisation; the analysis is the classical theory of the spherical harmonics on the three-sphere, and the operators of the group are the bi-invariant differential operators in the Casimir element.

A Nilpotent Group

Let $G$ be the Heisenberg group with its three-dimensional nilpotent algebra. The regular representation is not of finite type, its decomposition is by the unitary characters of the centre, and the derived representation of the algebra is faithful; the correspondence between the orbits of the coadjoint action and the irreducible unitary representations is Unitary Representations and the Orbit Method, below in the category.

Summary

The left and right regular representations of a Lie group are the actions on smooth functions by $L_gf(x) = f(g^{-1}x)$ and $R_gf(x) = f(xg)$; they are actions by algebra automorphisms, they commute, and the kernel of the two-sided action is the anti-diagonal copy of the centre. The derived representation of the Lie algebra is by the left-invariant fields, $X \mapsto \tilde X$, it is the derivative of the left regular representation, and it extends to an injective algebra homomorphism $U(\mathrm{G}) \to D_L(G)$ from the universal enveloping algebra onto the left-invariant differential operators; the right-invariant operators are the same statement with the opposite multiplication, and they commute with the left-invariant ones. The operators commuting with both actions are exactly the image of the centre $Z(U(\mathrm{G}))$, that is the bi-invariant operators; the group algebra acts by convolution and commutes with the left translations, while the enveloping algebra is realised as the distributions supported at the identity. On a compact group the Peter--Weyl theorem decomposes the regular representation into the irreducible unitary representations with multiplicity their dimension, and the Casimir operator, which is the Laplacian of a bi-invariant metric, acts on each summand by its quadratic invariant.

Summary of Notation

Symbol	Meaning
$L_gf(x) = f(g^{-1}x)$	the left regular representation
$R_gf(x) = f(xg)$	the right regular representation
$L_gR_h$	the two-sided action, with kernel the anti-diagonal centre
$\tilde X$	the left-invariant field, the derived representation of $X$
$U(\mathrm{G}) \to D_L(G)$	the extension to the enveloping algebra, injective
$\tilde v^{R}$	the right-invariant operator attached to $v$
$\tilde u\tilde v^{R} = \tilde v^{R}\tilde u$	commutation of the two derived representations
$\tilde Z(U(\mathrm{G}))$	the operators commuting with both actions
$k[G]$	the group algebra, acting by convolution
$\bigoplus_\pi V_\pi\otimes V_\pi^{*}$	the Peter--Weyl decomposition for compact $G$
$\langle\lambda,\lambda+2\varrho\rangle$	the Casimir eigenvalue on a summand