Background Material

0

Background Material

Introduction

The purpose of this chapter is to compile some of the background results, terminology and notation that will be used in this book. We recommend that the reader use this chapter basically for reference purposes. However, it might be worthwhile for the reader to skim through it on his first reading to become familiar with some of the notation and definitions. There are almost no proofs in this chapter. Everything covered can be found with adequate explanations in the references that we give, except for the material in Section 6. In Section 6 we give a noncommutative variant of the Artin-Rees Lemma of commutative algebra. There is a general Artin-Rees Lemma for nilpotent Lie algebras (see McConnell [11, Nouaze, Gabriel [ 13). Lemma 0.6.4appears for the first time in Stafford, Wallach [11. 0.1.

Invariant measures on homogeneous spaces

Let G be a locally compact topological group. Then a left invariant measure on G is a positive measure, dg, on G such that

0.1.1.

I

2

0.

Background Material

for all x E G and all f in (say) C,(G). If G is separable then it is well known (Haar’s theorem) that such a measure exists and that it is unique up to a multiplicative constant. If G is a Lie group with a finite number of components then a left invariant measure on G can be identified with a left invariant n-form on G (here dim G = n). If p is a non-zero left invariant n-form on G then the identification is implemented by integrating with respect to p using the standard method of differential geometry. If G is compact then we will (unless otherwise specified) use normalized left invariant measure. That is, the total measure is one. If dg is a left invariant measure and if x E G then we can define a new left invariant measure on G, p x , as follows: Px(f)

= j f(gx)dg. G

The uniqueness of left invariant measure implies that Px(f) =

6(x)

J f(g)dg.

G

with 6 a function of x which is usually called the modular function of G. If 6 is identically equal to 1 then we say that G is unimodular. If G is unimodular then we will call a left invariant measure (which is then automatically right invariant) inoariant. It is not hard to see that 6 is a continuous homomorphism of G into the multiplicative group of positive real numbers. This implies that if G is compact then G is unimodular. If G is a Lie group than the modular function of G is given by the following formula: 6 ( x ) = ldet Ad(x)l where Ad is the usual adjoint action of G on its Lie algebra. 0.1.2. Let M be a smooth manifold and let p be a volume form on M . Let G be a Lie group acting on M.Then ( g * p ) , = c ( g , x ) p xfor each g E G, x E M . One checks that c satisfies the cocycle relation

(1)

c ( g h , x )= c ( g , h x ) c ( h , x )

for h , g

E

G, x E M.

We will write J, f ( x )dx for J, fp. The usual change of variables formula implies that

for f (say) in C,(G) and g E G.

0.1. Invariant Measures on Homogeneous Spaces

3

Let H be a closed subgroup of G. We take M to be G / H . We assume that G has a finite number of connected components. A G-invariant measure, dx, on M is a measure such that

If dx comes from a volume form on M then (3) is the same as saying that Ic(g,x)l = 1 for all g E G, x E M . If M is a smooth manifold then it is well known that either M has a volume form or M has a double covering that admits a volume form. By lifting functions to the double covering (if necessary) one can integrate relative to a volume form on any manifold. Returning to the situation M = G / H , it is not hard to show that M admits a G-invariant measure if and only if the modular function of G restricted to H is equal to the modular function of H. Under this condition, a G-invariant measure on M is constructed as follows: let g be the Lie algebra of G and let b be the sub-algebra of g corresponding to H. Then we can identify the tangent space at 1H to M with g/b. The adjoint action of H on g induces an action Ad- of H on g/b. The above condition says that ldet Ad"(h)l = 1 for all h E H . Thus if H o is the identity component of H (as usual) and if p is a non-zero element of A"(g/Ij)* ( m = dim G / H ) one can translate p to a G invariant volume form on G / H o . Thus by lifting functions from M to G / H o one has a left invariant measure on M. Now Fubini's theorem says that we can normalize d g , dh and dx so that

(4) 0.1.3. Let G be a Lie group with a finite number of connected components. Let H be a closed subgroup of G and let dh be a choice of left invariant measure

on H. The following result is useful in the calculation of measures on homogeneous spaces.

Lemma. I f f is u continuous compactly supported function on H\G (note the

change to right cosets!)then there exists, g , a continuous compactly supported function on G such that f ( H x ) = J g(hx)dh. G

This result is usually proved using a "partition of unity" argument. For details see, for example, Wallach [ l , Chapter 23.

4

0. Background Material

Let G be a Lie group and let A and B be subgroups of G such that A n B is compact and that G = AB. The following result is useful for studying induced representations. 0.1.4.

Lemma. Assume that G is unimodular. If da is a left invariant measure on A and if db is a right invariant measure on B then we can choose an invariant measure, d g , on G such that J f ( g ) d g= J f ( a b ) d a d b AXE

G

forfEC,(G).

For a proof of this result see for example Bourbaki [l]. 0.2. The structure of reductive Lie algebras 0.2.1. Let g be a Lie algebra over C.We use the notation 3(9) for the center of g. Then g is said to be reductive if g = 3(g) 0 [g, g] with [g, g] semisimple. We recall the basic properties of g that will be used in this book with appropriate references. Recall that a subalgebra, lJ, of g is called a Cartan subalgebra if b is maximal subject to the conditions that Ij is abelian and if X E lJ then ad X is semisimple as an endomorphism of g. Here, if X , Y E g then ad X(Y) = [ X , Y] (as usual). Cartan subalgebras always exist and they are conjugate to one another under inner automorphisms (c.f. Jacobson [1. p.2731). If X E g then define the polynomials Djon g by det(t1 - ad X ) =

t’Dj(X),

here n = dim g . Let r be the smallest index such that 0,is not identically zero. Set D = D,. X E g is said to be regular if D ( X ) is nonzero.

Lemma. If X is regular then ad X is semi-simple. Futhermore, the centralizer in g of a regular element is a Cartan subalgebra of g (Jacobson [ l , p.591). Fix, 6, a Cartan subalgebra of g. If a E b* then we set ga = { X

E

g I [H,X ] = a ( H ) X

for all H

E

lJ>.

If a and ga are non-zero then we call a a root of g with respect to 6, and ga is called the root space corresponding to a. The set of all roots of g with respect to

5

0.2. The Structure of Reductive Lie Algehras

t, will be denoted @(g, 5) and called the root system of g (with respect to 5). We have

0 9,.

(1)

9=bO

(2)

If

(3)

I f a , B E ~ ( g , b ) t h e n l I 9 , , 9 / 1 1= % t o (Jacobson [1, p. 1 161).

(4)

ci

aE W g h )

E @(g,b)

then dim(g,)

=

1 (Jacobson [l, p.1111).

If a E @(g,6) then the only multiples of are c1 and - c1 (Jacobson [ 1, p. 1 161).

c1

in @(g, 6)

0.2.2. Let g be as above. If B is a symmetric bilinear form on g then B is said to be inoariant if B ( [ X , Y],Z)

=

-

B( Y, [ X , 21)

for all X , Y, Z

E

g.

A non-degenerate invariant form on g always exists. O n [g, g] one takes the Killing form Jacobson [l, p.691 and on j(g) one takes any non-degenerate symmetric form. The direct sum of the two forms is then a non-degenerate invariant form on g. Fix such a form, B. Fix a Cartan subalgebra, b, in g. It is clear that 5 is orthogonal, relative to B,to all of the’root spaces. We therefore see that (1) B restricted to t, is non-degenerate. Thus, if p E b* then we can define H , B(H, H,)

= p(H)

E

5 by for H E.)I

We can then define a non-degenerate symmetric bilinear form ( , ) on b* by b*. One has

( p , z) = B(H,, H , ) for p, z E

(2) (a,a) is a positive real number for c1 E @(g,f)). (Jacobson [l, p.1 lo]) Let bRdenote the real subspace of one has

b spanned by the H, for c1 E @(g,5). Then

(3) B restricted to bRis real valued and positive definite (Jacobson [l, p.1 IS]).

0.2.3. We retain the notation of the previous number. If ci E @(g,b) we denote by s, the reflection about the hyperplane CI = 0 in 6. That is, s,H = H - (2c1(H)/(c(,a))Ha

for H E 6.

0. Background Material

6

sa is called a Weyl rejection. The Weyl reflections have the following properties: (1)

sa@(g,6) = @(g,6)

(2)

sa6R = b R .

(Jacobson [l, p.1191).

We denote by W(g,9) the group generated by the Weyl reflections. W(g,6) is called the Weyl group of g with respect to 6. Let 6; denote the subset of all H E 6, such that a ( H ) is nonzero for all a E @(g,lj). Let C denote a connected component of 6;. Then C is called a Weyl chamber. (3) W(g,lj) acts simply transitively on the Weyl chambers (Bourbaki [2, p. 1633). A subset P of @(g,6) is called a system of positive roots if @(g, 5) is the disjoint union of P and - P ( = { - a I CI E P}) and if whenever a, p E P and a + p E @(g, 6) then a + p E P. If C is a Weyl chamber then the set of all a E @(g,6) that are positive on C is a system of positive roots. Conversely, if P is a system of positive roots then the subset of I-)R consisting of those H such that a ( H ) > 0 for all a E P is a Weyl chamber. Thus specifying a Weyl chamber is the same as specifying a system of positive roots. Fix a system of positive roots, P. Then a E Pis said to be simple if u cannot be written as a sum of two elements of P. The set of all simple roots of P is called a simple system for P or a basis for the root system @(g,$). Let 7t denote the simple system for P. Then z has the following properties (Jacobson [1, p.1201):

0.2.4.

(1)

7t is

a basis for (ljR)*.

(2) If p E P then fl

=

1 n,a

aEn

(3) W ( g ,6) is generated by the s,

with n, E N. for a E n

(Bourbaki [2, p.1551).

0.3. The structure of compact Lie groups

Let G be a compact Lie group with Lie algebra g. Let gc denote the complexification of g. Then gc is a reductive Lie algebra over C. In fact, if ( , ) is any positive non-degenerate symmetric bilinear form on g then we define a new form on g, ( , ), as follows:

0.3.1.

(X, y > = j (Ad(g)X, Ad(g)Y)dg G

for

x,y, E 9.

7

0.3. The Structure of Compact Lie Groups

Here (as usual) d g denotes normalized invariant measure on G. The invariance of dg immediately implies that (Ad(g)X, Ad(g)Y)

=

(X, Y )

for g

E

G and X, Y

E

g,

By differentiating this formula one sees that ( , ) is an invariant form on g. Thus, if u is an ideal of g then the orthogonal complement to u is also an ideal of 9. Hence, dimension considerations imply that g is a direct sum of 1dimensional and simple ideals. This clearly implies that g is reductive. Recall that the Killing form of g, B, is defined by the following formula:

B ( X , Y ) = t r ad X ad Y for X, Y E g. Since ad X is skew adjoint relative to ( , ) for X E g it is clear that B ( X , X) I 0 for X E 9. Also, B(X, X ) = 0 if and only if ad X = 0. Thus, g is semisimple if and only if B is negative definite. The converse is also true.

Theorem. ff g is a Lie algebra over R with negative dejinite Killing form then any connected Lie group with Lie algebra 9 i s compact. This theorem is known as Weyl’s theorem. For a proof see, for example, Helgason [1, Theorem 6.9, p. 1 331.

0.3.2. In this book a commutative compact, connected Lie group will be called a torus. Let T be a torus with Lie algebra t. If we look upon t as a Lie group under addition then exp is a covering homomorphism of t onto T. The kernel of exp is a lattice, L , in t. That is, L is a free Z module of rank equal to dim t. Let TAdenote the set of all continuous homomorphisms of T into the circle. If p E TAthen the differential of p (which we will also denote by p ) is a linear map of t into iR such that p ( L ) c 2niZ. If p is a linear map of t into iR such that p ( L ) c 2niZ then p is called integral. If p is an integral linear form on t then we define for t = exp(X), t w = exp(p(X)).This sets up an identification of integral linear forms on t and characters of T. 0.3.3. Let G be a compact, connected Lie group. Then a maximal torus of G is (as the name implies) a torus contained in G but not properly contained in any sub-torus of G. Fix a maximal torus, T, of G. Then t, is a Cartan subalgebra of 9., The elements of @(gc,),t are integral on t and thus define elements of T ” . Thus, we will look upon roots as characters of T. We now list some properties of maximal tori that will be used in this book.

8

0. Background Material

(1) A maximal torus of G is a maximal abelian subgroup of G (Helgason C1, p.2871). (2) If T and S are maximal tori of G then there exists an element g E G such that S = gTg-’ (Helgason [l, p.2481). (3) Every element of G is contained in a maximal torus of G. That is, the exponential map of G is surjective. (Helgason [l, p.1351.) (4)If T is a maximal torus of G then G / T is simply connected. (This follows from say Helgason [l, Cor.2.8, p.2871.)

Let T be a maximal torus of G. Let N ( T )denote the normalizer of T in G (the elements g of G such that gTg-’ = 7’). Let W(G,T ) denote the group N ( T ) / T .Then W(C,T ) is called the Weyl group of G with respect to T. If g E s E W(G,T ) then we set sH = Ad(g)H for H E t. This defines an action of W(G,T ) on t. ( 5 ) Under this action W(G,T ) = W(g,-,tc) (Helgason [l, Cor.2.13, p.2891).

0.3.4. Let g be a semisimple Lie algebra over C. Then a real form of g, u, will be called a compact form if 11 has a negative definite Killing form. The following result is due to Weyl. Combined with Theorem 0.3.1 it is the basis of what he called the “unitarian trick”.

Theorem. If b is a Cartan subalgebra of g then there exists a compact form, u, of g such that u n is maximal abelian in u. (Jacobson, [l, p.1471.) 0.4. The universal enveloping algebra of a Lie algebra

Let g be a Lie algebra over a field F which we will think of as R or C . Then a universal enveloping algebra for g is a pair ( A , j ) of an associative algebra with unit, 1, over F, A , and a Lie algebra homomorphism, j , of g into A (here an associative algebra is looked upon as a Lie algebra using the usual commutator bracket, [ X , Y] = X Y - Y X ) with the following universal mapping property: If B is an associative algebra with unit and if a is a Lie algebra homomorphism of g into B then there exists a unique associative algebra homomorphism CT- of A into B such that a(X) = o “ ( j ( X ) ) . It is easy to see that if ( A , j ) and (B, i) are universal enveloping algebras of g then there exists an isomorphism, T, of A onto B such that Tj = i. Thus, if a universal enveloping algebra exists then it is unique up to isomorphism. The usual construction of a universal enveloping algebra of g is given as follows: Let T ( g )denote the free associative algebra over F generated by the

0.4.1.

9

0.4. The Universal Enveloping Algebra of a Lie Algebra

vector space g. That is, T(g)is the tensor algebra over the vector space g. Let I ( g ) denote the two sided ideal of T ( g ) generated by the elements XY YX - [X, Y] for X, Y E g. Set U ( g ) = T(g)/Z(g). Let i denote the natural map of g into T(g).Let p denote the natural projection of T(g) into U ( g ) . Set j = pi. Then it is easy to see that ( U ( g ) , j )is a universal enveloping algebra for g. The basic result on universal enveloping algebras is the Poincare-BirkoffWitt Theorem (P-B-W for short):

Theorem. Let X , , . . . ,X,, be a basis of g. Then the monomials j ( X, )" . . . j(Xn)",

form a basis of U ( g )(Jacobson [l, p.1591). 0.4.2. In light of the uniqueness of universal enveloping algebras and P-B-W we will use the notation U ( g ) for the universal enveloping algebra of g and think of g as a Lie subalgebra of U ( g ) .Thus,j will be looked upon as the canonical inclusion. Let U m ( g denote ) the subspace of U ( g )spanned by the products of m or less elements of g. Then U m ( g )c U r n +'(9) defines a filtration of U ( g ) . This filtration is called the canonical jiltration of U ( g ) .With this filtration U ( g )is a filtered algebra (that is, U p ( g ) U 4 ( gc ) U p + q ( g ) ) . Let G r U ( g ) denote the corresponding graded algebra. g generates U ( g )and the elements XY - YX are in U ' ( g ) for X, Y E g. Hence Gr U ( g )is a commutative algebra over F. Let S(g) denote the symmetric algebra generated by the vector space g. Then there is a natural homomorphism, p, of S(g) onto Gr U(g).P-B-W implies that this homomorphism is an isomorphism. If X I , . . . ,X, are in g then set

symm (X, . . . X,)

= ( 1/ k !)

1X u

. . . Xu,

U

the sum over all permutations o of k letters. Then symm extends to a linear map of S(g) to U(g).Let q be the projection of Um(g)into G r U(g).If x E S(g) is homogeneous of degree k, then it is easily checked that q(symm(x)) = x. Hence symm defines a linear isomorphism of S(g) onto U(g).In particular, if X E g then symm(X") = X" (the multiplication on the left hand side is in S(g) on the right hand side it is in U ( g ) ) .symm defines a linear isomorphism of S(g) onto U ( g )which is called the symmetrization mapping. We note that if a the Lie algebra (0) then U(a) = F . Let E be the Lie algebra homomorphism of g onto a given by E(X)= 0. Then E extends to a homomorphism of U ( g )onto F which we also denote by E (rather than 6 " ) . E is called the augmentation homomorphism.

10

0. Background Material

We denote by goPpthe Lie algebra whose underlying vector space is g with bracket operation { X , Y } = [ Y , X ] . Then U(goPp)= U(g)Opp(the opposite algebra). The correspondence X H - X defines a homomorphism of g onto gOPP whose extension to U ( g )will be denoted x T . We note that the linear map x H x T is defined by the following three properties: (1)

l T = 1.

(2)

XT=-X

(3)

( ~ y=)y T~x T

for X E g. for x, y E U(g).

0.4.3. Let b be a subalgebra of g. P-B-W implies that the canonical map of U(b) into U ( g ) is injective. We can thus identify U(b) with the associative subalgebra of U ( g )generated by 1 and b. Let V be a subspace of g such that g = b @ V. Then P-B-W implies that the linear map U(b) 0 S ( V )

+

U(g)

Given by b 0 u H b symm(u) for b E U(b), v E S ( V ) , is a surjective linear isomorphism. Hence U ( g )is the free module on the generators symm(S(V ) )as a U(b) module under left multiplication. Similarly, U ( g )is the free right U(b) module generated by symm(S(V ) )under right multiplication by U(b).

0.5. Some basic representation theory 0.5.1. One of the most useful elementary results in representation theory is Schur’s Lemma. There is a Schur’s Lemma for most representation theoretic contexts (algebraic, unitary, Banach, etc.) In this book there will be several such Lemmas. We begin this section with a particularly useful one (usually called Dixmier’s Lemma). It is based on the following result: Lemma. Let V be a countable dimensional vector space ouer C . If T is an endomorphism of V then there exists a scalar c such that T - c l is not invertible on V. Suppose that T - cl is invertible for all scalars, c. Then P(T) is invertible on V for all non-zero polynomials P i n one variable. Thus if R = P / Q is a rational function with P and Q polynomials then we can define R ( T )to by the formula P(T)(Q(T)-’). This rule defines a linear map of the rational functions in one variable, C(x),into End(V). If u E V is non-zero and if R E C ( x ) is non-zero with R = P / Q as above then R(T)u = 0 only if P(T)u = 0. Thus the map of

0.5. Some Basic Representation Theory

C(x) into V given by R

11

R(T)u is injective. Since C(x)is of uncountable dimension over C this is a contradiction. H

0.5.2. We now come to Dixmier’s Lemma. Let V be a vector space over C. Let S be a subset of End( V ) .Then S is said to act irreducibly if whenever W is a subspace of V such that SW W then W = V or W = (0).

Lemma. Suppose that I/ is countable dimensional and that S c End(V) acts irreducibly, If T E End( V ) commutes with every element of S then T is a scalar multiple of the identity operator.

By 0.5.1 there exists c E C such that T - c l is not invertible on V. Since the elements of S preserve Ker(T - c l ) and Im(T - c l ) and since at least one of the two spaces must be proper, we see that T = c l . 0.5.3. Let g be a Lie algebra over F = R or C . Then a representation of g is a pair (a,V ) with V a vector space over C and a a homomorphism of g into

End(V). The universal mapping property of U ( g )implies that it extends to a representation of U(g).We will write (T rather than a- for this extension. If (T is understood we will usually use module notation for representations of Lie algebras (and their extensions to enveloping algebras). That is, we will write xu for (T(x)u.We will then call V a g-module or a U(g)-module (which, of course, it is in the usual associative algebra sense). If V and W are g-modules we denote by Hom,(V, W ) the space of all gmodule homomorphisms (or intertwining operators) from V to W. That is, the space of all linear maps, T, of V to W such that TXu = XTu for X E g and u E I/. We say that V and W are equivalent if there exists an invertible element in Horn,( V, W ) . Let V be a g-module. Then a subspace, W,of V is said to be inuariant if X W is contained in W for all X E g. V is said to be irreducible if the only invariant subspaces of V are V are (0). In this context Schur’s Lemma says: Lemma.

If V is an irreducible g-module then Horn,( V, V ) = CZ.

Let u be a non-zero element of V. Then U(g)u is an invariant non-zero subspace of V. Hence U(g)u = V. P-B-W (0.4.1) implies that U ( g )is countable dimensional. Thus V is a countable dimensional. The result now follows from Lemma 0.5.2. We now concentrate on a particularly important class of Lie algebras. A Lie algebra 5 over C is called a three dimensional simple Lie algebra (TDS for

0.5.4.

12

0. Background Material

short) if it has a basis H , X , Y with commutation relations [ X , Y ] = H , [ H , X ] = 2 X , [ H , Y ] = - 2Y. A concrete example of a TDS is eI(2, C)the Lie algebra of 2 by 2 trace zero matrices. Here one takes

x=["

'1,

Y = [ l0 0 O].

0 0

.=[;

3

We therefore see that if 5 is a TDS and if u is the real subalgebra of 5 with basis X - Y, i(X Y ) ,iH then u is isomorphic with the Lie algebra of SU(2) (the group of 2 by 2 unitary matrices of determinant 1). Let (a, V )be a finite dimensional representation of 5 (that is, dim V is finite). Since SU(2) is simply connected, there is a Lie homomorphism a- of SU(2) into GL( V )(the group of invertible elements of End(V)) whose differential is a restricted to u. Let du be normalized invariant measure on SU(2). Fix ( , ) a positive non-degenerate Hermitian form (inner product for short) on V. Then we define a new inner product ( , ) on V as follows:

+

(u, w ) =

J

SU(2)

(a-(u)u, a " ( u ) w ) du

for u, w E V.

Then ( ~ - ( U ) V , ( T " ( U ) W ) = ( u , w ) for u E S U ( 2 ) and u, w E V. Differentiating this relation gives ( X u , w ) = - ( u , X w ) for X E u and u, w E V. Thus if W is a 5-invariant subspace of V then so is the orthogonal complement of W. We have proved:

Lemma. I f V is a ,finite dimensional s-module then V splits into a direct sum of irreducible 5-submodules. The proof we have just used is a special instance of the celebrated "unitarian trick". This trick was also used in 0.3.1. 0.5.5. Thus to describe finite dimensional 5-modules it is enough to describe irreducible ones. To do this we will use the following commutation relation in U(5):

(1)

[ X , y"]= nY"-'(H - n

+ 1)

for n

=

1,2,

Let V be a finite dimensional irreducible s-module. Then H has an eigenvalue on V of maximal real part, c. Let u be a non-zero eigenvector for H with eigenvalue c. By the commutation relations defining a TDS we see that HXu = (c + 2)Xu. Thus Xu = 0. O n the other hand, (2)

HY"u = (c - 2n)Y"u

and X Y " u = n(c - n

+ l)Y"-'u

by (1). We therefore see that there must be a non-negative integer, m, such that

13

0.6. Modules Over the Universal Enveloping Algebra

Y"v is non-zero but Y m + l v= 0. Set vo = u and u, = Ynvfor n = 1,2,. ... Then ( 2 ) implies that v o , . . ., v, is a basis for a non-zero invariant subspace of V. Since V is irreducible, this implies that v o , . . . , v, is a basis of V. (2)now implies that tr H = ( m + l)(c - m) on I/. Since [ X , Y ] = H we must have tr H = 0 on V. Thus c = m. If W is an m + 1 dimensional vector space over C with basis w o , . . . , w,. We define the endomorphisms x, y and h of W by the following formulas:

( 3 ) xwo = 0, yw,=wntl hw,

xw,

= n(m - n

+ l ) ~ , - ~ for n = 1,..., m;

f o r n = O , ..., m - 1

= ( m - 2n)w,

for n

and

yw,=O;

= 0,. . ., m.

Then it is not hard to show that x, y, h satisfy the commutation relations of a TDS. Putting all of this together we have proved: Lemma. Let B be a TDS with standard basis X , Y, H. Then for every strictly positive integer m + 1 there exists up to equivalence exactly one irreducible m 1 dimensional irreducible s-module, W. Furthermore, W has a basis wo,..., w, such that X , Y, H correspond to the elements x , y, h in ( 3 ) respectively.

+

0.6. Modules over the universal enveloping algebra Let A be an associative algebra over C . Then A is said to be (left) Noetherian if whenever I , c . . . c 1, c ... is a chain of left ideals in A then there exists, m, such that 1, = 1, for all k > m. Let g be a Lie algebra over C .

0.6.1.

Lemma.

U ( g ) is Noetherian.

If I is a subspace of U ( g )set

Here the notation is as in 0.4.2.If 1 is a left ideal of U ( g )then Gr(l)is easily seen to be an ideal in Gr U ( g ) .Gr U ( g )is isomorphic with S(g). The Hilbert basis theorem implies that S(g)is Noetherian (Atiyah, Macdonald [l, p.811). Hence we conclude that there is m such that G r 1, = Gr 1, for all k > m. But then I,,,= lk for all k > m.

14

0. Background Material

0.6.2. If A is an algebra with unit over C then an A-module, M , is said to be finitely generated if there exist elements m,, . . ., m, of M such that M = E Amj.

Lemma. Let A be Noetherian and let M be a finitely generated A-module. I f M I c . . . c M,, c . . . is a chain of submodules of M then there exists m such that M,,, = Mk for all k > m. This is proved by induction on the number of generators and is left to the reader (cf Atiyah, Macdonald [ 1, p.751).

0.6.3. Let A be as in the previous Lemma. Let I be a two-sided ideal of A. We set I kequal to the ideal in A generated by the products of k elements of I. Then I is said to have the Artin-Rees property (AR property for short) if whenever M is a finitely generated A-module and N is a submodule of M there is a nonnegative integer k such that ( I k f j M )n

(1)

N

=

I j ( l k MnN )

for all j > 0.

If t is an indeterminate set A r t ] = A 0 C [ t ] .That is, A [ t ] is the algebra of all polynomials in t with coefficients in A. If I is a two sided ideal in A then we set I * = A + t l + t Z I 2+ ... + t k l k+ ... in A [ t ] .

Lemma.

I has the A R property if I* is a Noetherian algebra.

Let M be a finitely generated A-module. Set M*

=

M

+ t l M + t 2 1 Z M + ....

Then M * is a finitely generated I*-module. Let N be a submodule of M . Put N,

=N

Nk = N

+ t ( l M n N ) + t Z I ( I Mn N ) + ... + t'l'(1M

nN )

+ ...

+ t ( l M n N ) + . . . + t k ( l k Mn N ) + t k + ' l ( l k Mn N ) + ...

Then N , c N , c . . . is a chain of I *-submodules of M *. There is thus a k such that N k + j= Nk for all j > 0. This is the AR property. If n is a Lie algebra over a field then we set n, = [ n , n ] and n,, = [n,, n ] for m = 1, 2 , . . . . n is said to be nilpotent if there exists k such that nk = 0. Let g be a Lie algebra over C. Let n be a nilpotent Lie subalgebra of U ( g ) such that if X is in g then [ X , 111 c 11. Let I = nU(g). Then I is a two sided ideal in U ( g ) . 0.6.4.

0.6. Modules Over the Universal Enteloping Algehra

15

Proposition. I has the A R p r o p e r t y in U ( g ) . Setg" = g + t n + t 2 n , +t'ilt, + . . . i n U ( g ) [ t ] . S i n c e i i j = O f o r j > > O , g A i s a finite dimensional Lie algebra over C. Thus if i is the natural inclusion of g A into U ( g ) [ t ]then we have the extension i- to U ( g A )I.t is easy to check that i " ( U ( g " ) = I*. Thus since U ( g " )is Noetherian, I* is also. Thus Lemma 0.6.3 implies the result. 0.6.5. We conclude this section with a particularly important construction of U(g)-modules. Let b be a Lie subalgebra of g. Let M be a U (b)-module. Let U ( g )act on U(g) 0M by left translation in the first factor. Let V, be the U ( g ) submodule of U ( g ) @ M generated by the elements h 0m - 10 bm for m E M and b E U (b). Then we sct U h ) @ M = ( U ( g )0W l V M U(b)

We now collect some properties of this construction. Let N be a U ( g ) module and let T be a U (6)-module homomorphism of M into N , then (1) Then there exists a unique U(!J)-module homomorphism of U ( g )@, into N , T" such that T"(1 0m ) = Tm.

M

Indeed, put T - ( g 0m) = yT(m). Then Ker T - contains V,. Hence T induces a U(g)-module homomorphism T Aof U ( g )@,,, M into N . The rest is equally clear. (2) Let 0 + A 5 B

k C + 0 be a U (b)-module exact sequence. Then

is a U(g)-module exact sequence Let I/ be a subspace of g such that LJ = b 0 V. U ( g )= S(V )0 U ( 6) as a right U(6)-module under right multiplication (0.4.3).Thus we can look upon the modules U ( g )@,, D as S ( V )0D for D = A , B,C . Under this identification, a" = l o x

the result is now clear.

and

=lob,