<\body> <\expand|make-title> <\expand|make-title> A is constructed analogously to a smooth manifold. We specify an atlas (U,\)}>, where each chart M> is an open set and :U\\> is a homeomorphism. It is assumed that the transition functions \\:\(U\V)\\(U\V)> are holomorphic, for any two charts )> and )>. A is a Hausdorff topological group which is a complex manifold, in which the multiplication and inversion maps G\G> and G> are holomorphic. The Lie algebra of a complex Lie group is a complex Lie algebra. For example, )> is a complex Lie group. \; If > is a Lie algebra and \>, we say that and if We call the Lie algebra > if for all \>. <\proposition> The Lie algebra of an abelian Lie group is abelian. <\proof> The action of on itself by conjugation is trivial, so the induced action of on its Lie algebra is trivial. By Proposition 26 of it follows that End(Lie(G))> is the zero map, so . <\proposition> If is a Lie group, and and are commuting elements of , then =e e>. In particular, e=e e>. <\proof> First note that since the differential of is (Proposition 26 of ) ) Y=Y> for all . Recalling that )> is the endomorphism of induced by conjugation, this means that conjugation by > takes the one-parameter subgroup e> to itself, so e e=e>. Thus \ > and > commute for all real and . We recall from Section 4 of that the path > is characterized by the fact that >, while >(d/d t) = Y>. The latter condition means that if C>(G)> we have <\expand|equation*> f(p(t))=(Y f)(p(t)). Let e>. The vector field is invariant under left translation, in particular left translation by >, so <\expand|equation*> |\u>f(q(t,u))=(Y f)(ee). Similarly (making use of e=ee>),\ <\expand|equation*> |\t>f(q(t,u))=(X f)(ee). Now by the chain rule, <\expand|equation*> f(q(v,v))= |\t>f(q(t,u))\|+ |\u>f(q(t,u))\|= (Y f+X f)(q(v,v)). This means that the path r(v)=q(v,v)> satisfies >(d/d v)=(X+Y)> whence =ee>. Taking the result is proved. A is a compact connected Lie group which is abelian. For example, ={z\\> \|z\| =1}> is a torus. This group is isomorphic to /\>. Even though > and > are additive groups, we may, during the following discussion, sometimes write the group law in /\> multiplicatively. <\proposition> Let be a torus, and let > be its Lie algebra. Then \T> is a homomorphism, and its kernel is a lattice. We have (\/\)\\>, where is the dimension of . <\proof> Let > be the Lie algebra of . Since is abelian, so is >, and by Proposition , is a homomorphism from the additive group > to . The kernel \\> is discrete since is a local homeomorphism, and > is cocompact since is compact. Thus > is a lattice and \/\\(\/\)\\>. We may determine the complex representations of /\)> as follows. A character of > of the form <\equation> (x,\,x)\e i(kx)> where ,\,k)\\> induces a character on /\)>. \ and by classical Fourier analysis these characters span (/\)>)>>.\ <\proposition> Every irreducible complex representation of /\)> coincides with () for suitable \\>. <\proof> By classical Fourier analysis these characters span (/\)>)>>. It follows from the Peter-Weyl theorem that every complex representation decomposes into a direct sum of such characters. We also want to know the irreducible representations of /\)>. Let ,\, k\\> be given. Assume that they are not all zero. The complex character () is not a real representation. However, regarding it as a homomorphism /\)\\>, we may compose it with the real representation \t=ei \>\\)>|\)>>|\)>|\)>>>>>> of >. We obtain a real representation <\equation> (x,\,x)\ \ kx)>| \ kx)>>| \ kx)>|\ kx)>>>>>. <\proposition> Let /\)> and let (,V)> be an irreducible real representation. Then either > is trivial, or > is two dimensional, and is one of the irreducible representations (). <\proof> It is straightforward to see that the real representation () is irreducible. The completeness of this set of irreducible real representations follows from the corresponding classification of the irreducible complex characters (Proposition ). If is a compact torus, we will associate with a Lie group >>, which we call the of . Let >=\\\> be the complexification of the Lie algebra, and let >=\>/\>, where \\> is the kernel of \T>. We call this Lie group the of . It is easy to see that this construction is functorial: given a homomorphism :T\U> of compact tori, the differential >:Lie(T)\Lie(U)> commutes with the exponential map, so >> kills the kernel > of \T>. Therefore there is an induced map >\U>>.\ If we identify /\)>, the complexification >\(\/\)>. Since ei x>> induces an isomorphism of the additive group /\> with the multiplicative group >>, we see that >\(\>)>. We call any complex Lie group isomorphic to >)> a . By a > of a compact torus , we mean a continuous homomorphism \>>. These are just the characters of irreducible representations, known explicitly by (). They take values in >, as we may see from (1), or by noting that the image is a compact subgroup of >>. By a >> of a complex torus we mean an analytic homomorphism \>>.\ <\proposition> Let be a compact torus. Then any linear character > of extends uniquely to a rational character of >>. <\proof> Without loss of generality we may assume that /\)> and that >=(\>)>, where the embedding T>> is the map ,\,x)\(ei x>,\, e i x>)>. Every linear character of is given by () for suitable \\>, and this extends to the rational character ,\,t)\t>> of >>. Since a rational character is holomorphic, it is determined by its values on the image > of . We will denote the group of characters of a compact torus as >(T)>. We will denote its group law : if > and > are characters, then +\)(t)=\(t)\(t)>. We may identify >(*T)> with the group of rational characters of >>. A of a compact torus is an element such that the smallest closed subgroup of containing is itself. <\theorem> Let ,\,t)\\>, and let be the image of this point in /\)>. Then is a generator if and only if \ ,\,t> are linearly independent over >>>. <\proof> Let be the closure of the group t> generated by in /\)>. Then is a compact abelian group, and if it is not reduced to the identity it has a character >. We may regard this as a character of which is trivial on , and as such it has the form () for suitable \\>. Since itself is in , this means that k t\\>, so ,\,t> are linearly dependent. The existence of nontrivial characters of is thus equivalent to the linear dependence of ,\,t> and the result follows. <\corollary> Every compact torus has a generator. Indeed, generators are dense in . <\proof> We may assume that /\).> By Kronecker's theorem, what we must show is that -tuples ,\,t)> such that ,\,t> are linearly independent over > are dense in >. If ,\,t> are linearly independent, then linear independence of ,\,t> excludes only countably many >, and the result follows from the uncountability of >. <\proposition> Let /\)>. \; (i) Every automorphism of is of the form M t> (mod )>, where GL(r,\)>. Thus GL(r,\)>.\ \; (ii) If is a connected set and Aut(T)> is a continuous map such that f(h) t> is a continuous map T\T>, then is constant. We can express (ii) by saying that is discrete, since if it is given the discrete topology, then f(h) t> is continuous if and only if is continuous. <\proof> If : \ T\T> is an automorphism, then > induces an invertible linear transformation of the Lie algebra > of which commutes with the exponential map. It must preserve the kernel > of \T>. We may identify =\> in such a way that > is identified with >, in which case the matrix of must lie in )>. Part (i) is now clear. For part (ii), since is compact and continuous, as h>, f*(h)t> uniformly for T>. It is easy to see from (i) that this is impossible unless is locally constant. If is a group and a subgroup, we will denote by (H)> and (H)> the normalizer and centralizers of . If no confusion is possible we will denote these as simply and . Let be a compact, connected Lie group. It contains tori, for example , and an ascending chain \T\T\\> has length bounded by the dimension of . Therefore contains maximal tori. Let be a maximal torus. The normalizer G \| g T g=T}>. It is a closed subgroup, since if T> is a generator, is the inverse image of under the continuous map g t g>. <\proposition> N(T) is a closed subgroup of . The connected component >> of the identity in is itself. The quotient is a finite group. <\proof> We have a homomorphism Aut(T)> in which the action is by conjugation. By Proposition , GL(r,\)> is discrete, so any connected group of automorphisms must act trivially. Thus if N(T)>>, commutes with . If >\T>, then it contains a one-parameter subgroup \t\n(t)>, and and the closure of the group generated by and is a closed commutative subgroup strictly larger than . By the maximality of , it follows that >>. The quotient group >/T> is both discrete and compact, hence finite. The quotient is called the of with respect to . <\example> Suppose that . A maximal torus is <\expand|equation*> T=>||>||>|>|||>>>>> \ \|t\| = \ \ =\|t\| =1. Its normalizer consists of all monomial matrices (matrices with a single nonzero entry in each row and column) so the quotient S>. <\proposition> Let be a maximal torus in the compact, connected Lie group , and let >, > be the Lie algebras of and respectively. Then =\\\>, where > is invariant under . Under the restriction of to , > decomposes into a direct sum of two dimensional irreducible representations of , of the form (). <\proof> Since is compact, there exists a positive definite symmetric bilinear form on the real vector space > invariant under the real representation GL(\)>. The orthogonal complement > of > is invariant under (T). We claim that there are no invariant vectors in >. Indeed, if \> is such a vector, then by Proposition , is a one parameter subgroup not contained in which commutes with , and the closure of the group it generates with will be a torus strictly larger than , which is a contradiction. Since every nontrivial irreducible real representation of is of the form (), the result follows. <\corollary> If is a compact connected Lie group and a maximal torus, then is even.\ <\proof> This follows since )>, and > decomposes as a direct sum of two dimensional irreducible representations.\ We review the notion of an orientation. Let be a manifold of dimension . The of is a certain two fold cover which we now describe. One way of constructing > begins with the -fold exterior power of the tangent bundle: the fiber over M> is T(M)>. This is a one-dimensional real vector space. Omitting the origin and dividing by the equivalence relation w> if w> for \\\>, when , are elements of T(M)>>>, produces a set with two points. The disjoint union =M>F(x)> is topologized as follows. Let :\M> be the map sending to . If ,\,X> are vector fields which are linearly independent on an open set , then \\\X> determines, for each U>, an element of (x)>. We topologize > by requiring that > be a local homeomorphism.\ Now an of the manifold is a of the orientation bundle, that is, a continuous map > such that s(x)=x> for all M>. If an orientation exists, then > is a trivial cover, and \ M\(\/2\)>. In this case the bundle is called Any complex manifold is orientable. On the other hand a Möbius strip is not orientable. <\proposition> Let be a connected Lie group, a connected closed Lie subgroup. Then the quotient space is a connected, orientable manifold. <\proof> To make a manifold, choose a subspace > of =Lie(G)> complementary to =Lie(H)>. Then exp(X) g H> is a local homeomorphism of a neighborhood of the identity in > with a neighhood of the coset in . To see that is orientable, let :\M> be the orientation bundle, and let > be an element of (H)>. If G> then acts by left translation on , hence induces an automorphism > of >. We wish to define a global section of > by (\)>. We check that this is well defined. If H>, then g\H>, and we can find a path H> with > and g>. Let be the space of diffeomorphisms of , given the topology of uniform convergence, and consider the path Aut(M)> where M> is the map . We will follow Adams () in using Weil's 1935 topological proof of the conjugacy of maximal tori. This is based on the Lefschetz fixed point theorem, which we recall. We will prove a form of the Lefschetz fixed point formula suitable for our application. For a more general version, see Dold, , Chapter VII Section 6, and ``Fixed point index and fixed point theorem for Euclidean neighborhood retracts,'' (1965), 1--8. An alternative topological approach to the conjugacy of maximal tori is taken in Bröcker and Tom Dieck's . Suppose that is a manifold of dimension and M> a map. We define the of to be <\expand|equation*> \(f)=(-1) tr(f\|H(M,\)). A of is a solution to the equation . An isolated fixed point has a numerical multiplicity which we will denote (f)>. We will not define it in general. However if (M)\T(M)> is the differential of , then <\expand|equation*> I(f)=|0,>>|| 0,>>>>> with no conclusion if is not invertible. \; <\theorem> Assume that is compact and that has only isolated fixed points. Then <\expand|equation*> \(f)=I(f). We will assume this for the moment, then prove a weaker statement in the next section sufficient for the application. <\proposition> Let be a compact connected Lie group, and let be a maximal torus in . Let . Then is even-dimensional and orientable. If G>, let :X\X> be left translation by , so (x T)= g x T>. The Lefschetz number of > is equal to the order of the Weyl group . <\proof> The Lefschetz number clearly depends only on the homotopy class of , so to calculate it, we may take to be a generator > of . We note that is a fixed point of >> if and only if x T= x T>, that is, tx\T>. This is true if and only if N(T>), so the number of fixed points equals . We will show that (f>)=1> for each W>.\ We note that if W> (so normalizes then y T x=y x T> is a homeomorphism of with itself which commutes with >>, and this map takes x T>. So the multiplicities of the fixed points and are equal, and to compute (f>)> we may assume that . To compute (f>)> we notice that >(x T)= tx T=tx t>T>. We have a commutative diagram: <\expand|equation*> |>|>| F>>||f>>>||>|>>>> where >> is conjugation by >. Considering the differential at , we see that the differential of >:G/T\G/T> at is induced by )>. If > and > are the Lie algebras of > and >, we therefore have a commutative diagram: <\expand|equation*> \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ >|>| \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ >|>|(G/T)>>| Ad(t)>|| Ad(t)>|| d f>>>| \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ >|>| \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ >|>|(G/T)>>>>> It follows from Proposition that as a linear transformation, >> is equivalent to the image of > in a direct sum of irreducible two dimensional real representations of , which have the form () for various integer values of >. By Kronecker's Theorem, kt\\> for each of these, so >)> is a product of numbers of the form <\expand|equation*> det \kt)>| \kt)>>| \kt)>| \kt)>>>>>=\|1-e i \kt>\|\0, so (f>)=1>. As noted, this implies that (f>)=1> for every fixed point, and the result follows. <\theorem> Let be a compact connected Lie group, and let be a maximal torus. Then every maximal torus is conjugate to , and every element of is contained in a conjugate of . <\proof> Let G>. We will prove that is conjugate to an element of . This will prove that all maximal tori are conjugate to , for if > is another torus, we can choose to be a generator of >. Let and consider the map :X\X> be left translation by . It is sufficient to show that has a fixed point, since if then g x\T>. The Lefschetz number of > is by Proposition , so has fixed points. Since we do not prove the Lefschetz formula, the proof is not self-contained. This circumstance is mitigated by the fact that the result can be confirmed directly for particular Lie groups. For example when it is a consequence of the spectral theorem that any element can be conjugated to a diagonal element, and that any commutative family of unitary matrices can be simultaneously conjugated to diagonal matrices. So the theorem is true in this case, and similarly for the orthogonal groups. Still, one may be dissatisfied by this state of affairs. So in the next section we will give more self-contained topological arguments reproving Proposition . We recall that a space is if it is homeomorphic to a simplicial complex, in which case its singular homology is equal to the simplicial homology of the complex. This well known fact follows from Corollary 8.5 in Chapter V of Dold, , or Chapter III of Eilenberg and Steenrod, . <\proposition> Let be a Hausdorff topological space, and let M> be an automorphism of finite order without fixed points such that the quotient of by the action of is triangulable. Then the Lefschetz number of is zero. <\proof> Triangulating the quotient of by the action of , then pulling this triangulation back to , we obtain a triangulation > of which is invariant under . We may now compute the rational homology of using simplicial homology. The rational simplicial homology is the homology of a complex <\expand|equation*> 0\C\>C\>C\>\ where the > are finite dimensional vector spaces over >, and =0> if dim(M)>. Each > is the free vector space on those -simplices in the triangulation. Moreover, acts on the complex in each dimension by permuting these simplices, and no simplex is fixed by . So the trace of on > is zero in every dimension. Now let =ker(d)> and =im(d)>, so the homology (M)\Z/B>. We have a commutative diagram with exact rows: <\expand|equation*> ||>|>|>|>|>|>|>|>|||f>||f>||f>||>||>|>|>|>|>|>|>|>>>> Thus: <\equation> tr(f\|Z)=tr(f\|B)+tr(f\|H). Similarly, we have \ <\expand|equation*> ||>|>|>|>|>|>|>|>|||f>||f>||f>||>||>|>|>|>|>|>|>|>>>> Thus <\with|mode|math> <\equation> 0=tr(f\|C)=tr(f\|Z)+tr(f\|B). Now <\expand|equation*> \(f)=(-1) tr(f\|H)=(-1)tr(f\|Z)-(-1)tr(f\|B) =(-1)tr(f\|Z)+(-1)tr(f\|B)=0 by () and (). We now reconsider Proposition . Let ,\,P> be the fixed points of on . We know that in a neighborhood of a fixed point >, there exists a chart in which coordinates acts by a direct sum of linear actions of the form (). This means that the action of is linear near > in these coordinates and and T> maps \x\R(t) x>, where (t)> is the matrix <\equation> \*(t))>| \(t))>>| \(t))>| \*(t))>>>>>>>>>>>||>||>|>||| \*(t))>| \(t))>>| \(t))>| \*(t))>>>>>>>>>>>>>>> . Here ,\,\> are nonzero homomorphisms \/\> and . In this coordinate system, +x>\\>, +x>\\, \> defines an open polydisk > around > which is stable under . Denoting :X\X> left translation by G>, as in the proof of Proposition , the Lefschetz number of > is constant and we may choose at our convenience in calculating it. Instead of a generator of as in he proof of Proposition we will take \T> to be an element of finite order such that the (t)\\>. Since we may approximate a generator by a point of finite order, and since the > are the only fixed points of a generator, we may arrange so that >> has only the > as fixed points. We will apply Proposition to U>. By construction, >> no fixed points on , and so the Lefschetz number of >> on is zero. Now we consider the exact sequence of the pair : <\expand|equation*> \ \H(X,M)\H(M)\H(X)\H(X,M)\0. The alternating sum of the traces of >> on these homology groups is zero, and since the Lefschetz number of is zero, this means <\expand|equation*> \(f>)=(-1) tr(f>\|H(X,M)). We may compute this by excision. Let > be slightly larger polydiscs around the >, and let > be the interior of V>. Thus > is an open subset of and <\expand|equation*> \(f>)=(-1) tr(f>\|H(X-\,M-\)). Now the pair ,M-\)> consists of disconnected pieces, namely the pairs >, \ -U)>. Topologically -U> is a hollow shell, and the inclusion ,S)\(>, \ -U)> is a homotopy equivalence, so this piece has homology <\expand|equation*> H(>, \ -U)\>|>||>>>> Moreover the action of >> on this piece is homotopy equivalent to the identity, so\ <\expand|equation*> \(f>)=(-1) tr(f>\|H(>, \ -U))=1=\|W\|. In this section there will be few proofs, and nothing will be needed in the sequel. This section is included for cultural purposes only. <\proposition> The Euler characteristic (-1)dim H(G/T)> equals . <\proof> Take in Proposition . Quite a bit more can be proven: has nonzero cohomology only in even dimensions, so >(X)> is actually a commutative graded ring of dimension equal to . We've shown that is even dimensional. It is, in fact, a complex algebraic variety, a whose cohomology ring is at the center of an important crossroads of representation theory, combinatorics, topology and algebraic geometry. Proofs of some of this may be found in the last chapters of Fulton, To see why has a complex structure, let us take for definiteness. Let >=GL(n,\)>, and let be the of upper triangular matrices in )>. <\proposition> If >=GL(n,\)>, and is the standard Borel subgroup of >, then >=B G>=GB <\proof> If G>>, then applying the Gram-Schmidt orthogonalization matrix to the rows of , we may find an upper triangular matrix such that the rows of are orthonormal, that is, G>. It follows similarly that >=G B>. <\proposition> We have G>/B>>. <\proof> It follows from Proposition that every coset of >/B>> has a representative in , unique up to right multiplication by an element of >\G=T>. These facts, established for , extend to an arbitrary compact Lie group. We now see why has a complex structure.\ <\proposition> >=W>B w B> (disjoint). We will give a proof of this important fact later in the course. The double cosets , called give a cell decomposition of >>: one cell is open, the others lower dimensional, and the closure (w)=> of a cell is a union of the cell with lower dimensional Bruhat cells. The subspaces (w)> are called . They are algebraic subvarieties of , and their cohomology classes give a basis of >>. If is any field, we may consider , where is the group of upper triangular matrices in . Thus )>. If > is a finite field, the cardinality of )> is <\expand|equation*> )\||\|)\|>>>=-1)(q-q)\(q-q)|(q-1) q>=>-1|q-1>. This is a Gaussian ``factorial'', so called because a polynomial in whose value when equals . We denote this polynomial as >. For example, <\expand|equation*> (3!)=(q+q+1)(q+1)=q+2 q+2q+1. >(X)>>. That is, if is, as before, the complex flag manifold when , we find that\ <\expand|equation*> dim H(X)\>|>|>|>>>> so that dim H(X) q=(3!)>. At first, the fact that there is a relationship between the complex cohomology of a complex variety and the cardinality of the corresponding variety over a finite field seems quite mysterious. As another example of this phenomenon, let us consider the cohomology ring of the Grassmannian. If , let =U(n)/(*U(p)*\U(r))>. Like the flag manifold, it has a complex structure. We can represent it as =X(\)=GL(n,\)/P(\)>, where now for any field , is the ``parabolic'' subgroup consisting of matrices <\expand|equation*> |>||>>>> \ A\GL(p,F), D\GL(r,F). We may consider (\)=GL(n,\)/P(\)>. The order of this is the Gaussian binomial coefficient <\expand|equation*> =|(p!)(r!)> . Again, it is found that (\)> has onl even dimensional cohomology, and the the Gaussian binomial coefficient is a generating function for the dimensions of its cohomology groups. Motivated by these examples and other similar ones, Weil proposed a more precise relationship between the complex cohomology of a nonsingular projective variety and the number of solutions over a finite field. Proving the Weil conjectures required a new cohomology theory which was eventually supplied by Grothendieck. This is the -adic cohomology. Let >> be the algebraic closure of >, and let : X\X> be the geometric Frobenius map, which raises the coordinates of a point in to the -th power. The fixed points of > are then the elements of )>, and \ they may be counted by means of a generalization of the Lefschetz fixed point formula: <\expand|equation*> \|X(F)\|= (-1) tr(\\|H) \; The dimension of the -adic cohomology groups are the same as the complex cohomology, and in these examples (since all the cohomology comes from algebraic cycles) the odd dimensional cohomology vanishes and the trace of Frobenius on (X)> is >. Hence the Grothendieck-Lefschetz fixed point formula explains the extraordinary fact that the number of points over a finite field of the Grassmannian or flag varieties is a generating function for the complex cohomology. We are now in a position to appreciate Weil's comment at the beginning of his proof of the conjugacy of maximal tori () Weil asserts: \; <\quotation> L'analogie de ce théorème avec le célèbre théorème de Sylow sur les groupes finis est évidente, et sans doute profonde; aux groups d'ordre > de la théorie des groupes finis correspondent, s'il s'agit de Lie clos, les groupes abéliens (et sans doute les groupes intégrables dans le cas général). Voici une démonstration du théorème de Cartan qui suit de près la démonstration classique du théorème de Sylow. \; To see the relationship with the Sylow theorem, note that a Sylow subgroup of )> is the group )> of upper triangular unipotent matrices; the normalizer of )> is )>, so the coset space )> parametrizes the Sylow subgroups of , and the proof proceeds by showing an arbitrary -element has a fixed point in its action on this parameter space. More generally, for any linear algebraic group over a finite field, with Borel subgroup , the flag variety )=G(\)/B(\)> can be regarded as a parameter space for the set of Sylow subgroups. \; Let be a compact, connected Lie group, and let be a maximal torus. Theorem implies that every conjugacy class meets . Thus we should be able to compute the Haar integral of a class function (for example, the inner product of two characters) as an integral over the torus. The formula allowing this, the is therefore fundamental in representation theory, and in other areas too, such as random matrix theory. We have seen in Proposition that in the adjoint action on =Lie(G)>, restricted to , the Lie algebra > is an invariant subspace, complemented by a space > which decomposes as the direct sum of nontrivial two dimensional irreducible real representations as described in Proposition . Let be the Weyl group of . <\proposition> (i) Two elements of are conjugate in if and only if they are conjugate in . \; (ii) The inclusion G> induces a bijection between the orbits of on T and the conjugacy classes of . <\proof> Suppose that T> are conjugate in , say =u>. Let (u)>> be the connected component of the identity in the centralizer of . Then both and > are contained in , since they are connected commutative groups containing . As they are maximal tori in , they are maximal tori in , and so they are conjugate in the compact connected group . If H> such that =g T g> then g\N(T)>. Since =hu h=u>, we see that are conjugate in . (ii) is a restatement of (i). <\proposition> The centralizer . <\proof> Since N(T)>, is of finite index in . Thus if C(T)>, we have \T> for some . Let > be a generator of . Since the -th power map T> is surjective, there exists T> such that =t>. Now is contained in a maximal torus >, which contains > and hence T>. Therefore =T> and T>. The elements of the Weyl group are cosets for N(T)>. If T> the element > depends only on so by abuse of notation we denote it >. <\proposition> There exists a dense open set > of such that the elements > W)> are all distinct for \>. <\proof> If W>, let ={t\T\|w t w\t}>. It is an open subset of since its complement is evidently closed. If 1> and is a generator of , then \>, because otherwise if N(T)> represents , C(T)> so T> by Proposition , a contradiction since 1>. By Kronecker's theorem > is a dense open set. The finite intersection =1>\> thus fits our requirements. <\theorem> If is a class function, and if and are Haar measures on and (normalized so that and have volume ) then <\expand|equation*> f(g) d g=f(t) det([Ad(t)-I>] \| \) d t . <\proof> Let . We give the measure > invariant under left translation by such that has volume . Consider the map: <\expand|equation*> \:X\T\G,\(x T,t)=x t x. Both T> and are orientable manifolds of the same dimension.\ We compute the Jacobian > of >. Parametrize a neighborhood of by a chart based on a neighborhood of the origin in >. This chart is the map: <\expand|equation*> \\U\gx e T We also make use of the exponential map to parametrize a neighborhood of T>. This is the chart \V\t e>. We therefore have the chart near the point in T> which maps <\expand|equation*> \\\ \ \ \ (U,V)\ (x e T,t e) and in these coordinates, > is the map <\expand|equation*> (U,V)\ x et e ex. To compute the Jacobian of this map, we translate on the left by x> and on the right by . There is no harm in this because these maps are Haar isometries. We are reduced to computing the Jacobian of the map <\expand|equation*> (U,V)\ tet e e=e)U> e e. Identifying the tangent space of the real vector space \\> with itself, that is, with =\\\>, the differential of this map is <\expand|equation*> U+V\(Ad(t)-I>) U+V. The Jacobian is the determinant of the differential, so <\expand|equation*> (J\)(x T, t)=)-I>]\|\)>.> By Proposition , the map : X\T\G> is a -fold cover over a dense open set, and so for any function on , we have <\expand|equation*> f(g) d g=T>f(\(x T, t)) J(\(x T,t)) \ d x\dt. The integrand (x T, t)) J(\(x T,t))=f(t))-I>]\|\)>> > is independent of since is a class function, and the result follows. \; <\initial> <\collection> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|toc> |math font series||1Tori> |math font series||2The Conjugacy of Maximal Tori> |math font series||3Fixed Point Theory made Simple> |math font series||4Flag manifolds> |math font series||5The Weyl Integration Formula>