i (H) 1 on the diagonal and W acts as Sn t on by permuting a j.)

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1 1 Introduction Let G be a comact connected Lie Grou with Lie algebra g. T a maximal torus of G with Lie Algebra t. Let W = N G (T )/T be the Weyl grou of T in G. W acts on t through the Ad reresentations. W is generated by reflections across kernels of roots of t in g C or if you like the ositive real roots. The main result of these notes is that H(G/T ) vanishes in odd degrees. We will in fact rovide a ring isomorhism H(G/T ) to a urely algebraic structure. 2 Background/Review Let, be the Ad-invariant inner roduct on g (average all inner roducts on g or take the negative of the Killing Form). We then have an orthogonal decomosition g = m t. For X, Y, Z g, theinner roduct satisfies [X, Y ],X + Y,[X, Z] = 0. Note that Ad(T ) has no nonzero invariant vectors in m and no nonzero element of m has zero bracket will all of t (by maximally of t as an abelian subalgebra). A generic element H 0 t is such that ex H 0 is a regular element of T (i.e. its owers are dense in T ). Note, H 0 t is regular iff Ad(G) centralizer is recisely Ad(T ). For the remainder of this text, we choose some articular generic element H 0 t Let m = m 1... m v be an orthogonal decomosition given by the real irreducible reresentation of T, which are 2 dimensional. For H t, the eigenvalues of Ad ex H on m i are {ex(± 1α i (H)}, where α i t. We let the set of ositive roots + = {α 1,...,α v } be the set of roots that take ositive values on our generic element H 0. Note that since W acts faithfully on t, its image in GL(t) is generated by reflections about the kernels of elements in +. Since m i are reserved by ad(t), we can choose an orthonormal basis {X i,x i+v } for m i such that 0 α the matrix for ad(h) mi, H t, is i (H). Then ad-invariance of the inner roduct give α i (H) 0 us H, [X i,x j ] = [X i,h],x j = [H, X i ],X j = α i (H)X i+v,x j for 1 i v, 1 j 2v. Above, the right hand can be nonzero only if j = i + v. Thus,ifj = i ± v, then [X i,x j ] m. For 1 i v, weleth i =[X i,x i+v ], which is Ad(T )-invariant so H i m and ad(h i )m i m i. The san of X i,x i+v,h i is a Lie subalgebra of g that is actually isomorhic to su(2). 3 Invariant Theory Let P = =0 P be the symmetric algebra on t (i.e. P =(t ) / where λ 1... λ λ σ(1)... λ σ() for σ S ). One can think of P as olynomials over R where the monomials are roducts of functionals on t. The adjoint action of W on t induces an action/reresentations of W on P by degree-reserving algebra automorhisms (for λ t and w W, the action is λ λ Ad(w 1 )). We will be interested in the W -invariant olynomials P W. Examle 1. For U(n), P W is generated by elementary symmetric olynomials. For U(n), t is the set of diagonal comlex matrices with a j 1 on the diagonal and W acts as Sn t on by ermuting a j.) Theorem 2. (Chevalley) The ring P W has algebraically indeendent homogeneous generators F 1,...,F l with P W = R[F 1,...,F l ], where l =dimt. (Recall: algebraically indeendent means that the homomohism R[X 1,...,X l ] R[F 1,...,F l ] given by X i F i is an isomorhism) The generators are numbered such that deg F 1,...,deg F l. We will call the numbers m i =degf i 1 the exonents of W acting on t. It is known that m m l = v and (1 + m 1 )...(1 + m l )= W. Examle 3. SU(n) them i s are 1,...,n 1 and for G 2 they are 1, 5. Note that for SU(n) you loose the generator it degree 1, which you had for U(n), because of linear deendence. For G 2, the Lie algebra of T is that of SU(3) but the action of W is extended by an inversion. 1

2 Let D be the ring of constant coefficient differential oerators on P. We can thing of D as the symmetric algebra S(t), where H t corresonds to the function on t given by evaluation at H (e.g. H (λ 1 λ 2 )=λ 1 (H)λ 2 + λ 2 (H)λ 1 or the directional derivative for the vector H). We have that W acts naturally on D (by it s action on S(t)) and we define the harmonic olynomials in P to be those annihilated by the W -invariant differential oerators H = {f P : D W f =0}. One can think of H as the solution to a set of differential equations. Let H = H P,thenH = H since a differential oerator is W invariant if and only if each homogeneous comonent in W invariant (think the action of W on S(t)). Note that the action of W on P reserves H (for g W, P, D D, we have that D(g ) =(g 1 D)()). Proosition 4. If J is the ideal generated by the elements of P W of ositive degree, then P = H J and multilication is a linear isomorhism H P W P. The former gives us that P/J is isomorhic to H as W modules (Note: they are in fact isomorhic to the regular reresentation of W ). The isomorhism H P W P imlies dim H t = 0 l (1 + t + t t mi )(wherel =dimt) i=1 which shows that dim H v = 1 and H = 0 for >v. This formula is deduced from dim P t = dim H t dim(p W P )t, dim P t =(1+t + t ) l 1 = (1 t) l and dim(p W P )t = l i=1 1 (1 t mi ) l The rimordial harmonic olynomial is Π = α + α H v. For U(n) this is the Vandermonde determinant i<j (x i x j ), which is transformed by the sign character by the action of S n. In general, W act like the sign character on the san of Π, where the sign character ε : W {±1} gives the arity of the number of reflections for each g W. Any other olynomial whose san is reserved by the action of the sign character vanishes on all root hyerlanes and so is divisible by Π. Thus Π generates H v as dim H v = 1. We may now state the theorem we will discuss at the end of this talk Theorem 5. (Borel) There is a degree-doubling W -equivariant ring isomorhism c : P/J H(G/T ). Consequently, H (2) H(G/T ), where the subscrit indicated degree doubling. 4 Invariant Differential Forms Let G act transitively on a manifold M (think M = G/T ). If τ g is the diffeomorhism given by g G, then a differential -form ω Ω (M) isg-invariant if τ g ω = ω for all g G. Since G acts transitively, such a form is determined by its value at one oint on M. Lemma 6. Every de Rham cohomology class of M is reresented by a G-invariant form and the comlex of G-invariant forms is reserved by the exterior derivative. Definition 7. We define Λ n as the set of all skew-symmetric multilinear mas ω : n... n R where the domain has terms. Proosition 8. The comlex {(Λ n ) K,δ} comutes H (M), where K is the stabilizer of a oint o M, g = r n with r the Lie algebra of K, andδ is defined below. 2

3 Proof. Identify M = G/K and note that T o (M) is naturally identified with n. Thus, an invariant form ω is determined by a skew-symmetrc multilinear ma ω = ω o : n... n R, that is ω Λ n. The invariance of ω under K imlies that ω is Ad(K) invariant. Conversely, any element ω (Λ n ) K determines a G invariant form ω by ω g o ((dτ g )X 1,...,(dτ g )X )=ω(x 1,...,X ), for X 1,...,X n T o (M) and g G. Thus,wemayidentifytheG-invariant -forms with (Λ n ) K. The exterior derivative then become δ :(Λ n ) K (Λ +1 n ) K given by δω(x 0,...,X )= 1 +1 ( 1) i+j ω([x i,x j ] n,x 0,..., ˆX i,..., ˆX j,...,x ). i<j Where [X i,x j ] n is the rojection of [X i,x j ] on n along r and ˆ means the term is omitted. By the Lemma, the comlex {(Λ n ) K,δ} comutes H (M). Examle 9. Define ω(x, Y, Z) =X, [Y,Z] then [ω] = 0 H 3 (G). In articular, S n is not a Lie grou for n>3. 5 Cohomology of Flag Manifolds We will use Morse Theory to show that the odd dimensional cohomology of G/T vanishes. We can further use this aroach to decomose the flag manifold G/T into cells. This is called the Bruhat Decomosition. This rocess will be the generalization of decomosing the S 2 = SU(2)/T into a 0-cell and a 2-cell. We will find a Morse function f on G/T. For a smooth manifold M, a morse function f : M R is a smooth function with non-singular Hessian H x f at each critical oint x. The function we find will be the analogue of the dot roduct of vectors on a 2-shere with the vector ointing to the north ole. The san of the gradient flow lines emanating from a critical oint will rovide us with a cell decomosition. For the shere the flow lines from the south ole give us the 2-cell and the north ole, which has no flow lines emanating, gives us the 0-cell. If f is a Morse function and x is critical oint, let λ(x) be the number of negative eigenvalues of H x f. Then the Morse olynomial is M t (f) = t λ(x) over the critical oints x of f. Theorem 10. For a morse function f; M R, we have that M t (f) i dim Hi (M)t i. Moreover, if the morse olynomial has no consecutive exonents, equality holds. To construct a Morse function on G/T, we take the regular element H 0 t that we chose for the ositive roots. Recall that the Ad(G) centralizer of H 0 is exactly Ad(T ), so we may view G/T g as the Ad(G) orbit of H 0 (analogous to S 2 R 3 ). We define f : G/T R by For X g, we can comute the vector field f(gt) =Ad(g)H 0,H 0. Xf(gT) = d ds f(ex(sx)gt) s=0= Ad(g)H 0, [H 0,X], where the last equality is given by ad invariance of the inner roduct. Since the centralizer of H 0 in g is exactly t as H 0 is regular, it follows that the image of ad(h 0 )ism. So gt is a critical oint of f if and only if Ad(g)H 0, m = 0. Therefore, Ad(g)H 0 t by the orthogonal decomosition of g. It follows that Ad(g)H 0 = Ad(w)H 0 for some w W and that wt, for w W, are recisely the critical oints of f. Let X 1,...,X 2v be the orthonormal basis for m we discussed earlier. Note that the differential of π : G G/T mas Ad(w)m = m isomorhically onto T wt (G/T ), so we may use our basis to comute 3

4 the Hessian at each oint wt. If h ij is the ij entry in H wt f, then using our identities for the inner roduct h ij (wt) = X i Xj f(wt) =[X i, Ad(w)H 0 ], [H 0,X j ] = α i (Ad(w)H 0 )α j (H 0 )X i±v,x j±v. Note that it follows that h ij = 0 for i = j and h ii (wt) = α i (Ad(w)H 0 )α i (H 0 ). Since H 0 is regular, then so is Ad(w)H 0 and therefore h ii (w) = 0 and H wt f is non singular. Thus, as dim m =2ν, the index λ(wt) istwicethenumberm(w) of ositive roots α such that H α(ad(w)h) (i.e. w 1 α) is again a ositive root. The Morse olynomial of f is then M t (f) = w W t2m(w). Since all the exonents of M f (t) are odd, M t (f) = i Hi (M)t i and it follows that H i (M) = 0 for i odd. In articular, i dim H2i (G/T )= W. The Schubert cell X w in the Bruhat Decomosition is the cell sanned by the flow lines of the gradient of f emanating from wt. The dimension of this cell is then the number of ositive eigenvalues of the H wt f, or, equivalently, twice the number of ositive roots that become negative under w 1 α. Note that W acts on G/T by w gt = gw 1 T, which gives us an action of W on H(G/T ). Since H(G/T ) vanishes in odd degrees, the Lefschetz number associated to w is equal to the trace of its action on H(G/T ). If w = 1, then it has not fixed oints so the Lefshetz number is zero. If w = 1, then the Lefshetz number is simly the Euler characteristic, which is W. Hence, the action is that of the regular reresentation, so H(G/T ) R[W ] as W -modules. We can now give the roof of our final result, which we restate here. Theorem 11. (Borel) There is a degree-doubling W -equivariant ring isomorhism c : P/J H(G/T ). Consequently, H (2) H(G/T ), where the subscrit indicated degree doubling. Proof. The idea is to describe H(G/T ) in terms of G-invariant differential forms. For each λ t,we extend λ to all of g by making it zero on m and define an Ad(T )-invariant 2-form on m by ω λ (X, Y )=λ([x, Y ]). We can identify ω λ with an honest G-invariant differential form ω λ as before. The action of W on G-invariant forms is given by its action on G/T. One can comute that w ω λ = ω w λ. Further, the Jacobi identity imlies that δω λ (X, Y, Z) = 1 3 ([[X, Z] m,y] [[X, Y ] m,z] [[Y,Z] m,x]) = 0. We let c(λ) =[ ω λ ] H 2 (G/T ) and extend it to degree-doubling ma c : P H(G/T ) which reserves the W -action on both sides. Since H(G/T ) is the regular reresentation of W,its W -invariants are 1-dimensional and can therefore only occur in H 0 (G/T ). Since c is W -equivariant, it follows that the kernel of c contains the ideal J. The rest of the roof deals with showing that J is exactly the kernel of c. To rove that ker c = J, it suffices to show that c is injective on H as P = H J. This is done by induction starting at the highest degree of 2ν and descending down. For degree 2ν it suffices to show that c(π), where Π is the rimordial harmonic olynomial, is non zero in H 2ν (G/T ). For each root α i +,wehaveelementx i,x i+ν that form a basis for m i such that [X i,x i+ν ]= H i im t. Recall that [X i,x j ] m is j = i + ν where 1 i ν. For each i, writeω i = ω αi. Then by definition c(π) = [ ω α1... ω αν ] and we can evaluate ω 1... ω ν (X 1,X 1+ν,...,X ν,x 2ν )= 1 = 1 σ S 2ν sgn(σ)ω 1 (X σ(1),x σ(1+ν) ) ω ν (X σ(ν),x σ(2ν) )= σ S 2ν sgn(σ)α 1 ([X σ(1),x σ(1+ν) ]) α ν ([X σ(ν),x σ(2ν) ]) 4

5 Since α i ([X σ(i),x σ(i+ν) ]) = 0 unless [X σ i, X σ(i+ν) ] m, the term for σ is zero unless σ ermutes the airs {i, i + ν}, and ossibly switches the order of members. Note that σ(σ) is minus one the number of switches, so it follows that ω 1... ω ν (X 1,X 1+ν,...,X ν,x 2ν )= = 2ν 2ν σ S ν α 1 (H σ(1) ) α ν (H σ(ν) )= σ S ν α 1 ([X σ(1),x σ(1)+ν ]) α ν ([X σ(ν),x σ(ν)+ν ]) = 2ν 1 ν Π where i is the derivation of P extending λ λ(h i ). Since the airing D P R given by (D, f) (Df)(0) is erfect, it follows that there is a degree ν differential oerator that airs non trivially with Π. Further, since an irreducible W -module can only air non trivially with its dual, and the self-dual character ε ocurs with multilicity one in D ν, afforded by 1 ν, it follows that 1 ν Π = 0 and c(π) = 0. We may now inductively assume that c : H k H 2k (G/T ) is injective for some k ν. Let V = H k 1 ker c. Note that V is reserved by W since c is W -equivariant. Since the sign character is absent from H k 1, there is a ossible root α such that the reflection s α along the associated hyerlane does not act like I on V. We can then decomose V = V + V according to the eigensaces of s α. If V = 0, then V + = 0 so we may take some f V +. Now c(αf) =c(α)c(f) = 0 and αf is in degree k, soαf J by assumtion. Let h 1,...,h W be a basis for H with h 1,...,h r s α -skew and the rest s α -invariant. By Chevalley s Theorem, we can write αf = i h iτ i,withτ i W -invariant of ositive degree. Since αf is s α -skew by construction, the sum only goes u to r. For i r, the olynomial h i must vanish on ker α and therefore h i = αh i for some h i P. Then it follows that f = r i=1 h i τ i J and f is harmonic. Thus, we must have that f = 0 and c is injective on H k 1. By induction, c is injective and since H(G/T ) vanishes in odd degree, the roof is comlete. 5