A closed linear group $G$ is a group of real or complex matrices that is topologically closed in a complex general linear group. Rotation groups, unitary groups, and special linear groups provide familiar examples. The linear Lie algebra $\mathfrak g$ of $G$ is the set of derivatives at 0 of all smooth curves $c(t)$ of matrices that lie in $G$ for all $t$ and are equal to the identity at $t=0$. The set of matrices $\mathfrak g$ is indeed a Lie algebra over $\mathbb R$.

The exponential of a square matrix is defined by the familiar power series of the exponential function. The exponential map enables one to compute explicitly the linear Lie algebra of each of the familiar examples. It turns out that the exponential map carries $\mathfrak g$ into $G$. From this fact one deduces the main result of the Introduction, that any closed linear group has a natural structure as a smooth manifold that makes the group into a Lie group.

A homomorphism $\pi$ between two closed linear groups $G$ and $H$ carries smooth curves through the identity in $G$ to smooth curves through the identity in $H$. The map on the derivatives at 0 of such curves is well defined as a Lie algebra homomorphism $d\pi$ between the linear Lie algebras of $G$ and $H$. The homomorphisms $\pi$ and $d\pi$ are related by the important identity $\pi\circ\exp=\exp\circ\, d\pi$. This identity is a quantitative version of the statement that the infinitesimal behavior of a homomorphism at the identity determines the homomorphism in a neighborhood of the identity.

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field.

The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras---the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras.

Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size $n$-by-$n$ may be related to complex matrices of size $2n$-by-$2n$.

The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of $\mathfrak s \mathfrak l(2,\mathbb C)$. There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones.

Sections 10 through 13 contain a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory is summarized, and the correspondence is made with the concrete theory of closed linear groups as in the Introduction. In addition these sections discuss the adjoint representation, covering groups, and complex Lie groups.

The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with $\mathbb R^1$, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

The theme of this chapter is an investigation of complex semisimple Lie algebras by a two-step process, first by passing from such a Lie algebra to a reduced abstract root system via a choice of Cartan subalgebra and then by passing from the root system to an abstract Cartan matrix and an abstract Dynkin diagram via a choice of an ordering.

The chapter begins by making explicit a certain amount of this structure for four infinite classes of classical complex semisimple Lie algebras. Then for a general finite-dimensional complex Lie algebra, it is proved that Cartan subalgebras exist and are unique up to conjugacy.

When the given Lie algebra is semisimple, the Cartan subalgebra is abelian. The adjoint action of the Cartan subalgebra on the given semisimple Lie algebra leads to a root-space decomposition of the given Lie algebra, and the set of roots forms a reduced abstract root system.

If a suitable ordering is imposed on the underlying vector space of an abstract root system, one can define simple roots as those positive roots that are not sums of positive roots. The simple roots form a particularly nice basis of the underlying vector space, and a Cartan matrix and Dynkin diagram may be defined in terms of them. The definitions of abstract Cartan matrix and abstract Dynkin diagram are arranged so as to include the matrix and diagram obtained from a root system.

Use of the Weyl group shows that the Cartan matrix and Dynkin diagram obtained from a root system by imposing a ordering are in fact independent of the ordering. Moreover, nonisomorphic reduced abstract root systems have distinct Cartan matrices. It is possible to classify the abstract Cartan matrices and then to see by a case-by-case argument that every abstract Cartan matrix arises from a reduced abstract root system. Consequently the correspondence between reduced abstract root systems and abstract Cartan matrices is one-one onto, up to isomorphism.

The correspondence between complex semisimple Lie algebras and reduced abstract root systems lies deeper. Apart from isomorphism, the correspondence does not depend upon the choice of Cartan subalgebra, as a consequence of the conjugacy of Cartan subalgebras proved earlier in the chapter. To examine the correspondence more closely, one first finds generators and relations for any complex semisimple Lie algebra. The Isomorphism Theorem then explains how much freedom there is in lifting an isomorphism between root systems to an isomorphism between complex semisimple Lie algebras. Finally the Existence Theorem says that every reduced abstract root system arises from some complex semisimple Lie algebra. Consequently the correspondence between complex semisimple Lie algebras and reduced abstract root systems is one-one onto, up to isomorphism.

For a complex Lie algebra $\mathfrak g$, the universal enveloping algebra $U(\mathfrak g)$ is an explicit complex associative algebra with identity having the property that any Lie algebra homomorphism of $\mathfrak g$ into an associative algebra $A$ with identity “extends” to an associative algebra homomorphism of $U(\mathfrak g)$ into $A$ and carrying 1 to 1. The algebra $U(\mathfrak g)$ is a quotient of the tensor algebra $T(\mathfrak g)$ and is a filtered algebra as a consequence of this property. The Poincaré-Birkhoff-Witt Theorem gives a vector-space basis of $U(\mathfrak g)$ in terms of an ordered basis of $\mathfrak g$.

One consequence of this theorem is to identify the associated graded algebra for $U(\mathfrak g)$ as canonically isomorphic to the symmetric algebra $S(\mathfrak g)$. This identification allows the construction of a vector-space isomorphism called “symmetrization” from $S(\mathfrak g)$ onto $U(\mathfrak g)$. When $\mathfrak g$ is a direct sum of subspaces, the symmetrization mapping exhibits $U(\mathfrak g)$ canonically as a tensor product.

Another consequence of the Poincaré-Birkhoff-Witt Theorem is the existence of a free Lie algebra on any set $X$. This is a Lie algebra $\mathfrak F$ with the property that any function from $X$ into a Lie algebra extends uniquely to a Lie algebra homomorphism of $\mathfrak F$ into the Lie algebra.

This chapter is about structure theory for compact Lie groups, and a certain amount of representation theory is needed for the development. The first section gives examples of group representations and shows how to construct new representations from old ones by using tensor products and the symmetric and exterior algebras.

In the abstract representation theory for compact groups, the basic result is Schur's Lemma, from which the Schur orthogonality relations follow. A deeper result is the Peter-Weyl Theorem, which guarantees a rich supply of irreducible representations. From the Peter-Weyl Theorem it follows that any compact Lie group can be realized as a group of real or complex matrices.

The Lie algebra of a compact Lie group admits an invariant inner product, and consequently such a Lie algebra is reductive. From Chapter I it is known that a reductive Lie algebra is always the direct sum of its center and its commutator subalgebra. In the case of the Lie algebra of a compact connected Lie group, the analytic subgroups corresponding to the center and the commutator subalgebra are closed. Consequently the structure theory of compact connected Lie groups in many respects reduces to the semisimple case.

If $T$ is a maximal torus of a compact connected Lie group $G$, then each element of $G$ is conjugate to a member of $T$. It follows that the exponential map for $G$ is onto and that the centralizer of a torus is always connected. The analytically defined Weyl group $W(G,T)$ is the quotient of the normalizer of $T$ by the centralizer of $T$, and it coincides with the Weyl group of the underlying root system.

Weyl's Theorem says that the fundamental group of a compact semisimple Lie group $G$ is finite. Hence the universal covering group of $G$ is compact.

In any finite-dimensional representation of a complex semisimple Lie algebra $\mathfrak g$, a Cartan subalgebra $\mathfrak h$ acts completely reducibly, the simultaneous eigenvalues being called “weights.” Once a positive system for the roots $\Delta^+(\mathfrak g,\mathfrak h)$ has been fixed, one can speak of highest weights. The Theorem of the Highest Weight says that irreducible finite-dimensional representations are characterized by their highest weights and that the highest weight can be any dominant algebraically integral linear functional on $\mathfrak h$. The hard step in the proof is the construction of an irreducible representation corresponding to a given dominant algebraically integral form. This step is carried out by using “Verma modules,” which are universal highest weight modules.

All finite-dimensional representations of $\mathfrak g$ are completely reducible. Consequently the nature of such a representation can be determined from the representation of $\mathfrak h$ in the space of “$\mathfrak n$ invariants.” The Harish-Chandra Isomorphism identifies the center of the universal enveloping algebra $U(\mathfrak g)$ with the Weyl-group invariant members of $U(\mathfrak h)$. The proof uses the complete reducibility of finite-dimensional representations of $\mathfrak g$.

The center of $U(\mathfrak g)$ acts by scalars in any irreducible representation of $\mathfrak g$, whether finite dimensional or infinite dimensional. The result is a homomorphism of the center into $\mathbb C$ and is known as the “infinitesimal character” of the representation. The Harish-Chandra Isomorphism makes it possible to parametrize all possible homomorphisms of the center into $\mathbb C$, thus to parametrize all possible infinitesimal characters. The parametrization is by the quotient of $\mathfrak h^*$ by the Weyl group.

The Weyl Character Formula attaches to each irreducible finite-dimensional representation a formal exponential sum corresponding to the character of the representation. The proof uses infinitesimal characters. The formula encodes the multiplicity of each weight, and this multiplicity is made explicit by the Kostant Multiplicity Formula. The formula encodes also the dimension of the representation, which is made explicit by the Weyl Dimension Formula.

Parabolic subalgebras provide a framework for generalizing the Theorem of the Highest Weight so that the Cartan subalgebra is replaced by a larger subalgebra called the “Levi factor” of the parabolic subalgebra.

The theory of finite-dimensional representations of complex semisimple Lie algebras has consequences for compact connected Lie groups. One of these is a formula for the order of the fundamental group. Another is a version of the Theorem of the Highest Weight that takes global properties of the group into account. The Weyl Character Formula becomes more explicit, giving an expression for the character of any irreducible representation when restricted to a maximal torus.

Every complex semisimple Lie algebra has a compact real form, as a consequence of a particular normalization of root vectors whose construction uses the Isomorphism Theorem of Chapter II. If $\mathfrak g_0$ is a real semisimple Lie algebra, then the use of a compact real form of $(\mathfrak g_0)^{\Bbb C}$ leads to the construction of a “Cartan involution” $\theta$ of $\mathfrak g_0$. This involution has the property that if $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$ is the corresponding eigenspace decomposition or “Cartan decomposition,” then $\mathfrak k_0\oplus i\mathfrak p_0$ is a compact real form of $(\mathfrak g_0)^{\Bbb C}$. Any two Cartan involutions of $\mathfrak g_0$ are conjugate by an inner automorphism. The Cartan decomposition generalizes the decomposition of a classical matrix Lie algebra into its skew-Hermitian and Hermitian parts.

If $G$ is a semisimple Lie group, then a Cartan decomposition $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$ of its Lie algebra leads to a global decomposition $G=K\exp\mathfrak p_0$, where $K$ is the analytic subgroup of $G$ with Lie algebra $\mathfrak k_0$. This global decomposition generalizes the polar decomposition of matrices. The group $K$ contains the center of $G$ and, if the center of $G$ is finite, is a maximal compact subgroup of $G$.

The Iwasawa decomposition $G=KAN$ exhibits closed subgroups $A$ and $N$ of $G$ such that $A$ is simply connected abelian, $N$ is simply connected nilpotent, $A$ normalizes $N$, and multiplication from $K\times A\times N$ to $G$ is a diffeomorphism onto. This decomposition generalizes the Gram-Schmidt orthogonalization process. Any two Iwasawa decompositions of $G$ are conjugate. The Lie algebra $\mathfrak a_0$ of $A$ may be taken to be any maximal abelian subspace of $\mathfrak p_0$, and the Lie algebra of $N$ is defined from a kind of root-space decomposition of $\mathfrak g_0$ with respect to $\mathfrak a_0$. The simultaneous eigenspaces are called “restricted roots,” and the restricted roots form an abstract root system. The Weyl group of this system coincides with the quotient of normalizer by centralizer of $\mathfrak a_0$ in $K$.

A Cartan subalgebra of $\mathfrak g_0$ is a subalgebra whose complexification is a Cartan subalgebra of $(\mathfrak g_0)^{\Bbb C}$. One Cartan subalgebra of $\mathfrak g_0$ is obtained by adjoining to the above $\mathfrak a_0$ a maximal abelian subspace of the centralizer of $\mathfrak a_0$ in $\mathfrak k_0$. This Cartan subalgebra is $\theta$ stable. Any Cartan subalgebra of $\mathfrak g_0$ is conjugate by an inner automorphism to a $\theta$ stable one, and the subalgebra built from $\mathfrak a_0$ as above is maximally noncompact among all $\theta$ stable Cartan subalgebras. Any two maximally noncompact Cartan subalgebras are conjugate, and so are any two maximally compact ones. Cayley transforms allow one to pass between any two $\theta$ stable Cartan subalgebras, up to conjugacy.

A Vogan diagram of $\mathfrak g_0$ superimposes certain information about the real form $\mathfrak g_0$ on the Dynkin diagram of $(\mathfrak g_0)^{\Bbb C}$. The extra information involves a maximally compact $\theta$ stable Cartan subalgebra and an allowable choice of a positive system of roots. The effect of $\theta$ on simple roots is labeled, and imaginary simple roots are painted if they are “noncompact,” left unpainted if they are “compact.” Such a diagram is not unique for $\mathfrak g_0$, but it determines $\mathfrak g_0$ up to isomorphism. Every diagram that looks formally like a Vogan diagram arises from some $\mathfrak g_0$.

Vogan diagrams lead quickly to a classification of all simple real Lie algebras, the only difficulty being eliminating the redundancy in the choice of positive system of roots. This difficulty is resolved by the Borel and de Siebenthal Theorem. Using a succession of Cayley transforms to pass from a maximally compact Cartan subalgebra to a maximally noncompact Cartan subalgebra, one readily identifies the restricted roots for each simple real Lie algebra.

The first main results are that simply connected compact semisimple Lie groups are in one-one correspondence with abstract Cartan matrices and their associated Dynkin diagrams and that the outer automorphisms of such a group correspond exactly to automorphisms of the Dynkin diagram. The remainder of the first section prepares for the definition of a reductive Lie group: A compact connected Lie group has a complexification that is unique up to holomorphic isomorphism. A semisimple Lie group of matrices is topologically closed and has finite center.

Reductive Lie groups $G$ are defined as 4-tuples $(G,K,\theta,B)$ satisfying certain compatibility conditions. Here $G$ is a Lie group, $K$ is a compact subgroup, $\theta$ is an involution of the Lie algebra $\mathfrak g_0$ of $G$, and $B$ is a bilinear form on $\mathfrak g_0$. Examples include semisimple Lie groups with finite center, any connected closed linear group closed under conjugate transpose, and the centralizer in a reductive group of a $\theta$ stable abelian subalgebra of the Lie algebra. The involution $\theta$, which is called the “Cartan involution” of the Lie algebra, is the differential of a global Cartan involution $\Theta$ of $G$. In terms of $\Theta$, $G$ has a global Cartan decomposition that generalizes the polar decomposition of matrices.

A number of properties of semisimple Lie groups with finite center generalize to reductive Lie groups. Among these are the conjugacy of the maximal abelian subspaces of the $-1$ eigenspace $\mathfrak p_0$ of $\theta$, the theory of restricted roots, the Iwasawa decomposition, and properties of Cartan subalgebras. The chapter addresses also some properties not discussed in Chapter VI, such as the $KA_{\frak p}K$ decomposition and the Bruhat decomposition. Here $A_{\frak p}$ is the analytic subgroup corresponding to a maximal abelian subspace of $\mathfrak p_0$.

The degree of disconnectedness of the subgroup $M_{\frak p}=Z_K(A_{\frak p})$ controls the disconnectedness of many other subgroups of $G$. The most complete description of $M_{\frak p}$ is in the case that $G$ has a complexification, and then serious results from Chapter V about representation theory play a decisive role.

Parabolic subgroups are closed subgroups containing a conjugate of $M_{\frak p}A_{\frak p}N_{\frak p}$. They are parametrized up to conjugacy by subsets of simple restricted roots. A Cartan subgroup is defined to be the centralizer of a Cartan subalgebra. It has only finitely many components, and each regular element of $G$ lies in one and only one Cartan subgroup of $G$. When $G$ has a complexification, the component structure of Cartan subgroups can be identified in terms of the elements that generate $M_{\frak p}$.

A reductive Lie group $G$ that is semisimple has the property that $G/K$ admits a complex structure with $G$ acting holomorphically if and only if the centralizer in $\mathfrak g_0$ of the center of the Lie algebra $\mathfrak k_0$ of $K$ is just $\mathfrak k_0$. In this case, $G/K$ may be realized as a bounded domain in some $\mathbb C^n$ by means of the Harish-Chandra decomposition. The proof of the Harish-Chandra decomposition uses facts about parabolic subgroups. The spaces $G/K$ of this kind may be classified easily by inspection of the classification of simple real Lie algebras in Chapter VI.

An $m$-dimensional manifold $M$ that is oriented admits a notion of integration $f\mapsto\int_Mf\omega$ for any smooth $m$ form. Here $f$ can be any continuous real-valued function of compact support. This notion of integration behaves in a predictable way under diffeomorphism. When $\omega$ satisfies a positivity condition relative to the orientation, the integration defines a measure on $M$. A smooth map $M\to N$ with $\dim M\lt\dim N$ carries $M$ to a set of measure zero.

For a Lie group $G$, a left Haar measure is a nonzero Borel measure invariant under left translations. Such a measure results from integration of $\omega$ if $M=G$ and if the form $\omega$ is positive and left invariant. A left Haar measure is unique up to a multiplicative constant. Left and right Haar measures are related by the modular function, which is given in terms of the adjoint representation of $G$ on its Lie algebra. A group is unimodular if its Haar measure is two-sided invariant. Unimodular Lie groups include those that are abelian or compact or semisimple or reductive or nilpotent.

When a Lie group $G$ has the property that almost every element is a product of elements of two closed subgroups $S$ and $T$ with compact intersection, then the left Haar measures on $G$, $S$, and $T$ are related. As a consequence, Haar measure on a reductive Lie group has a decomposition that mirrors the Iwasawa decomposition, and also Haar measure satisfies various relationships with the Haar measures of parabolic subgroups. These integration formulas lead to a theorem of Helgason that characterizes and parametrizes irreducible finite-dimensional representations of $G$ with a nonzero $K$ fixed vector.

The Weyl Integration Formula tells how to integrate over a compact connected Lie group by first integrating over conjugacy classes. It is a starting point for an analytic treatment of parts of representation theory for such groups. Harish-Chandra generalized the Weyl Integration Formula to reductive Lie groups that are not necessarily compact. The formula relies on properties of Cartan subgroups proved in Chapter VII.

The definition of unitary representation of a compact group extends to the case that the vector space is replaced by an infinite-dimensional Hilbert space, provided care is taken to incorporate a suitable notion of continuity. The theorem is that each unitary representation of a compact group splits as the orthogonal sum of finite-dimensional irreducible invariant subspaces. These invariant subspaces may be grouped according to the equivalence class of the irreducible representation, and there is an explicit formula for the orthogonal projection on the closure of the sum of all the spaces of a given type. As a result of this formula, one can speak of the multiplicity of each irreducible representation in the given representation.

The left-regular and right-regular representations of $G$ on $L^2(G)$ are examples. So is the left-regular representation of $G$ on $L^2(G/H)$ for any closed subgroup $H$. More generally, if $H$ is a closed subgroup and $\sigma$ is a unitary representation of $H$, the induced representation of $\sigma$ from $H$ to $G$ is an example. If $\sigma$ is irreducible, Frobenius reciprocity says that the multiplicity of any irreducible representation $\tau$ of $G$ in the induced representation equals the multiplicity of $\sigma$ in the restriction of $\tau$ to $H$.

Branching theorems give multiplicities of irreducible representations of $H$ in the restriction of irreducible representations of $G$. Three classical branching theorems deal with passing from $U(n)$ to $U(n-1)$, from $SO(n)$ to $SO(n-1)$, and from $Sp(n)$ to $Sp(n-1)$. These may all be derived from Kostant's Branching Theorem, which gives a formula for multiplicities when passing from a compact connected Lie group to a closed connected subgroup. Under a favorable hypothesis the Kostant formula expresses each multiplicity as an alternating sum of values of a certain partition function.

Some further branching theorems of interest are those for which $G/H$ is a compact symmetric space in the sense that $H$ is the identity component of the group of fixed elements under an involution of $G$. Helgason's Theorem from Chapter VIII translates into a theorem in this setting for the case of the trivial representation of $H$ by means of Riemannian duality. An important example of a compact symmetric space is $(G\times G)/\text{diag}\,G$; a branching theorem for this situation tells how the tensor product of two irreducible representations of $G$ decomposes.

A cancellation-free combinatorial algorithm for decomposing tensor products for the unitary group $U(n)$, due to Littlewood and Richardson, is of great utility. It leads to branching theorems for the compact symmetric spaces $U(n)/SO(n)$ and $U(2n)/Sp(n)$. In turn the first of these branching theorems help in understanding branching for the compact symmetric space $SO(n+m)/(SO(n)\times SO(m))$.

Iteration of branching theorems for compact symmetric spaces permits analysis of some complicated induced representations. Of special note is $L^2(K/(K\cap M_0))$ when $G$ is a reductive Lie group, $K$ is the fixed group under the global Cartan involution, and $MAN$ is the Langlands decomposition of any maximal parabolic subgroup.

If $G$ is a connected complex Lie group that is the complexification of a compact Lie group $U$, a “prehomogeneous vector space” for $G$ is a complex finite-dimensional vector $V$ together with a holomorphic representation of $G$ on $V$ such that $G$ has an open orbit in $V$. The open orbit is necessarily unique. Easy examples include the standard representation of $GL(n,\mathbb C)$ on $\mathbb C^n$, the standard representation of $Sp(n,\mathbb C)$ on $\mathbb C^{2n}$, the action of $K^{\mathbb C}$ on $\mathfrak p^+$ when $G/K$ is Hermitian, and certain actions obtained from the standard Vogan diagrams of some of the indefinite orthogonal groups.

The question that is to be studied is the decomposition of the symmetric algebra $S(V)$ under $U$. For any prehomogeneous vector space, the symmetric algebra $S(V)$ embeds in a natural $U$ equivariant fashion into $L^2(U/U_v)$, where $U_v$ is the subgroup of $U$ fixing a point $v$ in $V$ whose $G$ orbit is open. This fact gives a first limitation on what representations can occur in $S(V)$.

A “nilpotent element” $e$ in a finite-dimensional Lie algebra $\mathfrak g$ is an element for which $\text{ad}\,e$ is nilpotent. If $\mathfrak g$ is complex semisimple, the Jacobson–Morozov Theorem says that such an $e$, if nonzero, can be embedded in an “$\mathfrak s \mathfrak l_2$ triple” $(h,\,e,\, f)$, spanning a copy of $\mathfrak s \mathfrak l(2,\mathbb C)$.

When a complex semisimple Lie algebra is graded as $\bigoplus \mathfrak g^k$, $\text{ad}\,\mathfrak g^0$ provides a representation of $\mathfrak g^0$ on $\mathfrak g_1$, and Vinberg’s Theorem says that the result yields a prehomogeneous vector space. All such gradings arise from parabolic subalgebras of $\mathfrak g$. The examples above of the action of $K^{\mathbb C}$ on $\mathfrak f p^+$ and of certain actions obtained from indefinite orthogonal groups are prehomogeneous vector spaces of this kind.

For the first of these two examples, the action of $K$ on $S(\mathfrak p ^+)$ is described by a theorem of Schmid. In the special case of $SU(m,n)$, this theorem reduces to a classical theorem about the action of the product of two unitary groups on the space of polynomials on a matrix space. For the second of these two examples, the action on the symmetric algebra can be analyzed by using this classical theorem in combination with Littlewood’s Theorem about restricting representations from unitary groups to orthogonal groups.

In the general case of Vinberg’s Theorem, if $v$ is suitably chosen in the prehomogeneous vector space $V$, then $U/(U_v)_0$ fibers by a succession of three compact symmetric spaces, and hence $L^2(U/(U_v)_0)$ can be analyzed by iterating various branching theorems for compact symmetric spaces. This fact gives a second limitation on what representations can occur in $S(V)$.

If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Also any linear map of $E$ into $T(E)$ extends to a derivation of $T(E)$.

The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative.

Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1.

The tensor algebra, the symmetric algebra, and the exterior algebra are all examples of graded associative algebras. A more general notion than a graded algebra is that of a filtered algebra. A filtered associative algebra has an associated graded algebra. The notions of gradings and filtrations make sense in the context of vector spaces, and a linear map between filtered vector spaces that respects the filtration induces an associated graded map between the associated graded vector spaces. If the associated graded map is an isomorphism, then the original map is an isomorphism.

A ring with identity is left Noetherian if its left ideals satisfy the ascending chain condition. If a filtered algebra is given and if the associated graded algebra is left Noetherian, then the filtered algebra itself is left Noetherian.

A finite-dimensional real Lie algebra is the semidirect product of a semisimple subalgebra and the solvable radical, according to the Levi decomposition. As a consequence of this theorem and the correspondence between semidirect products of Lie algebras and semidirect products of simply connected analytic groups, every finite-dimensional real Lie algebra is the Lie algebra of an analytic group. This is Lie's Third Theorem.

Ado's Theorem says that every finite-dimensional real Lie algebra admits a one-one finite-dimensional representation on a complex vector space. This result sharpens Lie's Third Theorem, saying that every real Lie algebra is the Lie algebra of an analytic group of matrices.

The Campbell-Baker-Hausdorff Formula expresses the multiplication rule near the identity in an analytic group in terms of the linear operations and bracket multiplication within the Lie algebra. Thus it tells constructively how to pass from a finite-dimensional real Lie algebra to the multiplication rule for the corresponding analytic group in a neighborhood of the identity.

This appendix contains information about irreducible root systems, simple Lie algebras over $\mathbb C$ and $\mathbb R$, and Lie groups whose Lie algebras are simple, noncompact, and noncomplex. The first two sections deal with the root systems themselves and the corresponding complex simple Lie algebras. The last two sections deal with the simple real Lie algebras that are noncompact and noncomplex and with their corresponding Lie groups.