Calculation of Thom polynomials for minimal orbits

Diploma Thesis

Gergely Bérczi Eötvös University, Budapest

Supervisor: László Fehér

June 2003

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Contents

1. Introduction

The theory of Thom polynomials is part of an enormous area of algebraic topology called theory of obstructions. The word obstruction means ‘”obstacle to the existence of an object”. A great variety of problems can be translated into the following type of question: when has a fiber bundle a section? In this case obstructions are cohomology elements of the base space so that if they do not vanish, then this is an obstacle: the bundle will not have any section.

The Thom polynomials show up when our fiber bundle is a vector bundle, and η ⊂ V is a closed subset of the fiber, which is invariant under the structure group of the bundle. In this case it make sense to speak about η in an arbitrary fiber, and to ask whether there exist a section avoiding η at every point. There is a so called avoiding ideal of the cohomology ring of the base space, whose elements must vanish if the bundle admits such section, so elements of this ideal are obstructions, this ideal is an obstruction ideal. When the avoiding ideal has a generator, it is called Thom polynomial. Determination of the avoiding ideal is usually slightly easier problem than finding the generator of it, which is also called first obstruction By now theory of Thom polynomials has become more than a homotopy theoretic question; it connects several branches of mathematics, like singularity theory, representation theory, algebraic geometry, etc. For example the degeneracy loci formulas in algebraic geometry turned out to be Thom polynomials for group actions.

Calculation of Thom polynomials is not an easy task, despite the quite simple definition. By now there are three known efficient methods for calculating them. The first, classical way is the method of resolutions – this is also effective at computations of other invariants of singular varieties. The second, which turned out to be the most effective method is based on an idea of R. Rimányi and called the method of restriction equations. A standard reference of this is [?], and a nice summary of this method and its applications is [?]. This thesis deals with the third method, the so-called Duistermaat- Heckman formula, which is a well-known formula in algebraic topology. A special form of this formula makes us able to compute Thom polynomials for smooth invariant projective varieties.

Organization of the paper. This thesis can be divided into four parts. The first part consist of section §2, where we present the background and the notions of the theory. The second part (§3, §4, §5) deals with Lie groups and Lie algebras and gives a short summary of their representations. In §4 we present the basic facts about representations of the classical Lie groups, which we will use later in computations. In section §5 we introduce the subjects of our later examinations, namely the flag manifolds and minimal orbits: the new result of this thesis is the method of applying the Duistermaat-Heckman formula for these varieties. In the third part (§6) we prove the Duistermaat-Heckman formula from the integral formula of Atiyah and Bott, and we discuss the connection between the affine and projective Thom polynomial. In the last part (§7) we present the method of calculations of Thom polynomials for minimal orbits, and give some examples. The new result here is the Thom polynomials of the orthogonal grassmannians and the spinor varieties, see ??, ??.

Acknowledgments. I would like to thank my teachers for their work they invested in me. First of all Prof. András Szűcs, who has taught me the basics of algebraic and differential topology, and thanks for the interesting lectures he has given us. I am grateful to my supervisor László Fehér, and to Richárd Rimányi for the beautiful topic, and the constant inspiration for work. Thanks for Balázs Kőműves for the valuable discussions. Last but not least I would like to acknowledge the patience and support of my family and Bea.

2. Obstruction Theory, Theory of Thom polynomials

Let us start with some motivation and some words about the historical background and the origin of the theory.

2.1. Motivation. The name Thom polynomial comes from singularity theory, where R. Thom proposed the following question ([?])

Problem 2.1. Given a smooth map f : M → N consider the points of M where f has singularity of type η, this is denoted by η(f). What is the cohomology class [\overline{η(f)}] ∈ H^{∗}(M) defined via the Poincaré duality by the closure of η(f) ⊂ M?

The simplest case of this problem is the following: We have a smooth map f : M → N. What is the Poincaré dual of the closure of points in M where df, the Jacobian of f has corank k?

As Thom observed, this class can be expressed as a polynomial of characteristic classes of the vector bundles T M and f^{∗}T N. We can see easily the connection between this problem and that of finding a section of a vector bundle avoiding an invariant closed subset. Let us see the most simple case, the so called Σ_{k} singularities. The following problem is the same that of Thom, if we use it for E = T M, F = f^{∗}T N, and for the section df:

Problem 2.2. Let E and F be complex vector bundles of complex dimension n and p over the manifold M, and let s : E → F be a vector bundle homomorphism, that is, a section of the bundle HOM(E, F). Let Σ_{k}(s) denote the set of m ∈ M where the linear map s(m) has corank k. What is the cohomology class [\overline{Σ_{k}(s)}] ∈ H^{∗}(M), the Poincaré dual of the closure of Σ_{k}(s)?

Note, that \overline{Σ_{k}(s)} is just ∪_{j=k}^{n}Σ_{j}(s), since the closure of the points where the corank is k contains the points where the corank is greater than k. Actually, the class [\overline{Σ_{k}(s)}] is an obstruction for the existence of a section of the following fiber bundle: let we look at the subspace Σ_{<k} ⊂ Hom(ℂ^{n}, ℂ^{p}) containing matrices of corank smaller than k. Σ_{<k} is an invariant subset for the group G = GL(n, ℂ) × GL(p, ℂ) acting on Hom(ℂ^{n}, ℂ^{p}). So in each fiber F^{x}, (x ∈ M) of the bundle HOM(E, F) we can define Σ_{<k}^{x} ⊂ F^{x}. A section s with s(x) ∈ Σ_{<k}^{x} for every x ∈ M is the same as s(x) ∉ \overline{Σ}_{k}^{x} for every x ∈ M. If there is a section s avoiding ∪_{x∈M} \overline{Σ}_{k}^{x}, then—since s induces an isomorphism between the cohomology of the total space and base space—[\overline{Σ_{k}(s)}] ∈ H^{∗}(M) must be zero, so this Poincaré dual is an obstruction.

We give the precise definitions and background to these notions in the next section. The problem was solved by Porteous, here are the formulas ([?])

Theorem 2.3. (Porteous)[?]

[\overline{Σ_{k}(s)}] = det(A)

where A_{i,j} = c_{i−j+k} for i, j = 1, …, k + p − n and c_{l} = c_{l}(F − E) are the Chernclasses of the difference bundle F − E = f^{∗}T N − T M.

Our goal is to find cohomological obstructions of sections of fiber bundles, that is, cohomology classes of the base space which have to be zero if the fiber bundle admits a section.

2.2. The Borel construction, equivariant cohomology. Let F be an arbitrary topological space, G a Lie group, and ρ : G → Homeo(F) a (left) action of G on F with homeomorphisms (with diffeomorphisms if F is differentiable manifold, and in general the group of morphism of the given category) Then we call F → E → B a ρ-bundle, if the structure group is given by ρ. When F = V a vector space and ρ : G → GL(V), this is a vector bundle. When F = G is a Lie group, and ρ : G → Homeo(G) is the left multiplication this is called principal G-bundle. The total space of a principal G-bundle admits a free right G-action, because G acts on itself by right multiplication as well, and this commutes with the left multiplication. The factor by this free right action is the projection map of the bundle.

For a given bundle ξ : F_{ξ} → E_{ξ} → B_{ξ} and a map f : B → B_{ξ} we can define f^{∗}ξ, the pull back of ξ, which is a F_{ξ}-fiber bundle over B. Pull backs of homotopic maps f ∼ f′ : B → B_{ξ} are isomorphic. In a similar construction we get the associated bundle: if ξ : G → P → M is a principal G-bundle, and ρ : G → Homeo(F) is a left action, then the associated F -fiber bundle is ρ_{∗}ξ : F → E → M where

E = P ×_{ρ} F = (P × F)/(p, f) ∼ (pg^{−1}, gf)

Note that here we used the fact, that G acts from left on the fiber but on the right on the total space. We can do from a right action of G on a space X a left action by xg ⇝ g^{−1}x, so we can say, that we factored by the left diagonal action.

With association we can construct universal bundles. We call a ρ-bundle universal ρ-bundle, if every ρ-bundle is the pull back of this. If we want to get a universal ρ-bundle, first we have to build a universal principal G-bundle, which was done by Milnor:

Theorem 2.4. (Milnor) For every Lie group G there exist universal principal Gbundle G → EG → BG. The space EG is contractible with a free (right) G-action on it, and BG = EG/G. Every universal principal G-bundle can be obtained from a contractible space, which has a free right G-action on the total space, and the base space is the factor space. Moreover the space BG is unique up to homotopy equivalence if we require to be a CW -complex. This is called the classifying space of G.

If ξ : G → P → M is a principal G-bundle then by there is a map f : M → BG with f^{∗}(EG) = ξ, this is called classifying map. For example BGL(n) is the Grassmannians manifold. Now we get a universal ρ bundle where ρ : G → Homeo(F) with association: F → B_{G}F = EG ×_{ρ} F → BG is a universal ρ-bundle. Note that the classifying space mapping G → BG is a functor from the category of Lie groups to the category of classifying spaces: if H → G is a Lie group homomorphism then we can define a map Bf : BH → BG in the following way: G has a natural left H-action given by hg = f(h)g, and Bf is the classifying map of the principal G-bundle EH ×_{H} G → BH. Analogously, forming the universal ρ bundles is a functor too, introduced by Borel:

Borel-construction: For any G-space X we can construct B_{G}X = EG ×_{G} X, which is the total space of a bundle with fiber X over the classifying space BG associated to the universal bundle EG → BG. Note that wee will sometimes denote this Borel construction by B_{G}X = X_{G}. This is a covariant functor between the category of G-spaces with morphisms the G-equivariant maps, and the Borel-construction of the corresponding spaces, so a G-equivariant map f : X → Y induce a map Bf : B_{G}X → B_{G}Y, which is not well defined just the homotopy class of it, but if we fix the space BG, then Bf will be well-defined: it is the natural bundle map between B_{G}X → BG and B_{G}Y → BG over the same base space which is “the map f in each fiber”. Note, that BG = EG ×_{G} pt is the Borel construction of the point. Now

H_{G}^{∗}(X) = H^{∗}(B_{G}X)

is a G-equivariant cohomology theory, i.e. it is a contravariant functor from the category of G-spaces to graded rings. This is an extraordinary cohomology theory, that is, all axioms of cohomology is satisfied except for the axiom for the cohomology of a point, which can be rather complicated, as a theorem of Borel says, see Theorem ??.Observe, that the cohomology group ⊕H_{G}^{i}(X) is also a module over the ring H_{G}^{∗} = H_{G}^{∗}(pt) = H^{∗}(BG) where the module structure is induced by the projection π : BV → BG, so ασ := π^{∗}(α)σ for α ∈ H_{G}^{∗}. The usual properties of ordinary cohomology remain valid, so for a G-equivariant map X → Y we have H_{G}^{∗}(Y) → H_{G}^{∗}(X), we have exact sequences, excision, etc.

We have a natural map σ of B_{G}M onto the quotient space M/G, which fits into the following diagram of Cartan and Borel:

\xymatrix{EG \ar[d] & EG × M \ar[l] \ar[d] \ar[r] & M \ar[d] \\ BG & (EG ×_{G} M) \ar[l]^{π} \ar[r]^{σ} & M/G}

In general σ is not a fibering. In fact, it is easy to see that the fiber σ^{−1}(Gm) over the orbit Gm is the quotient of EG by the stabilizer G_{m} of m at the right action of G on EG. So σ^{−1}(Gm) is the classifying space of G_{m}:

B_{G}η = σ^{−1}(Gm) = EG/G_{m} ≃ BG_{m}

where η = Gm is the orbit of m in M, and trivially σ^{−1}(η) = B_{G}η. So we have H_{G}^{∗}(η) = H^{∗}(BG_{η}) for any orbit of M. Note that this equation includes only the orbit η, so we have

Theorem 2.5. If the Lie group G acts transitively from the left on the space η, then H_{G}^{∗}η ≅ H_{G_{x}}^{∗}pt = H^{∗}(BG_{x}) where G_{x} is the stabilizer group of arbitrary point of η.

If G is a compact Lie-group acting smoothly on M, the map σ induces homotopy equivalence whenever G acts freely, so that B_{G}M ≃ M/G in that case. So the map

σ^{∗} : H^{∗}(M/G) → H_{G}^{∗}(M)

is an isomorphism if G acts freely. In general, B_{G}M can be considered as the “homotopy theoretic quotient” of M by G. Let H be a closed subgroup of G, then G/H is an orbit at the natural action of G with stabilizer H, so we get as above:

H_{G}^{∗}(G/H) = H_{H}^{∗} = H^{∗}(BH).

There is a natural homomorphism

i^{∗} : H_{G}^{∗}(M) → H^{∗}(M)

from equivariant theory to ordinary theory which corresponds to the inclusion i : M ↪ B_{G}M of M as the fiber over a point of BG.

Although the equivariant theory is largely analogous to the ordinary theory, its overtones are rather different, precisely because the equivariant theory of a “point” is so large. Observe, that in the equivariant theory the “points” of M/G really correspond to orbits of G on M, and hence are of various “sizes”, according to the size of the stabilizer group. Thus the free orbits carry only zero-dimensional equivariant cohomology, while the point orbits stand at the other extreme and their cohomology is H_{G}^{∗}.

2.3. Obstruction ideals. The standard reference of this subsection is [?], we go more or less along this reference. Let us suppose that a Lie group G act on the fiber F, and F → E → M be a fiber bundle associated to a principal G-bundle G → P → M – i.e. E = P ×_{G} F. We are mainly interested in the case when G is a semisimple Lie group and F is a G-manifold.

Our starting point is the following well-known observation:

Observation 2.6. Sections of the fiber bundle E = P ×_{G} F are in one to one correspondence with G equivariant maps from P to F. (We have left action on F, right action on F, which can be transformed into a left action.)

In other words in the category of left G-spaces a section s of E is a lift of the

collapse map pt_{P} : P → pt: \xymatrix{ & F \ar[d]^{pt_{F}} \\ P \ar[r]_{pt_{P}} \ar[ur]^{s} & pt}

Suppose now that h is any contravariant functor from the category of (left) Gspaces into (graded) rings (the main example is G-equivariant cohomology). Then a section s induces a commutative diagram: \xymatrix{ & h(F) \ar[dl]_{h(s)} \\ h(P) & h(pt) \ar[l]^{h(pt_{P})} \ar[u]_{h(pt_{F})}}

So the existence of the section s implies that

ker h(pt_{F}) ⊂ ker h(pt_{P}). (1)

We will use the following notation: O_{F} = ker h(pt_{F}) is the obstruction ideal of F. Now we can reformulate (??) in the following way:

Theorem 2.7. If the fiber bundle E = P ×_{G} F admits a section then all characteristic classes of P from the obstruction ideal O_{F} vanish.

So the goal of a general theory would be the calculation of O_{F} for G-spaces F and G-equivariant cohomology theories, and from now on we deal with the introduced cohomology H_{G}. Observe, that B_{G}P = EG×_{G}P ≅ P/_{r}G = M, since the right action of G on the total space of a principal G-bundle is free. Moreover, Bpt_{P} : M → BG is the classifying map of the principal G-bundle P → M. So after using the Borel construction we get from the diagram above:

\xymatrix{ & B_{G}F \ar[d]^{Bpt_{F}} \\ B_{G}P = M \ar[r]_{Bpt_{P}} \ar[ur]^{Bs} & Bpt = BG}

and

\xymatrix{ & H_{G}^{∗}(F) \ar[dl]_{H_{G}(s)} \\ H_{G}^{∗}(P) = H^{∗}(M) & H_{G}^{∗}(pt) = H^{∗}(BG) \ar[l]^{H_{G}(pt_{P})} \ar[u]_{H_{G}(pt_{F})}}

So in this case O_{F} ⊂ H^{∗}(BG). Suppose now, that F is d − 2 connected, and π_{d−1}(F) ≅ ℤ. Then if we have a triangulation of M with small simplices (so the bundle over every simplex is trivial) then we have a section over the d − 1-skeleton, and the first obstruction sits in H^{d}(M, π_{d−1}(F)) = H^{d}(M, ℤ). The vanishing of the first obstruction is a sufficient and necessary condition for the existence of a section over the d-skeleton of the bundle. For the universal F -bundle EG ×_{G} F → BG, the first obstruction is an element of H^{d}(BG, ℤ). Let we denote the first obstruction of the bundle E by fo(E) and that of the universal bundle by fo(F). If k is the classifying map of E, that is

\xymatrix{E = P ×_{G} F \ar[r]^{K} \ar[d]^{F} & B_{G}F \ar[d]^{F} \\ M \ar[r]^{k} & BG}

then k^{∗}fo(F) = fo(E) (that is, the first obstruction is natural). So it is enough to define it for the universal case, and we do it after Fuchs, Fomenko and Gutenmacher ( see [?]) with the aid of the spectral sequence of the bundle B_{G}F → BG. In the E_{2} table of this E_{2}^{p,0} = H^{p}(BG), E_{2}^{p,q} = 0 for 1 ≤ q ≤ d−2 (since F is d−2-connected), and E_{2}^{0,d−1} = H^{d−1}(F) = π_{d−1}(F) = ℤ. Let us denote the boundary homomorphism from E_{∗}^{0,d−1} to E_{∗}^{d,0} by f. We have a natural homomorphism e_{p} : E_{2}^{p,0} → E_{∞}^{p,0} → H^{p}(B_{G}F) which is called edge homomorphism. Now the edge homomorphism e_{d} is quite simple: the kernel of it is just the image of f, so Im e_{d} = ker f. It can be seen easily that e_{d} = π_{d}^{∗} where π_{d}^{∗} : H^{d}(BG) → H^{d}(B_{G}F) is induced by the projection. Let 1 ∈ ℤ = E_{2}^{0,d−1} be the generator, then we define the first obstruction as fo(F) := f(1) ∈ H^{d}(BG). It can be proved, that it has the properties described above. Since fo(F) is the generator of ker π_{d}^{∗} = O_{F} ∩H^{d}(BG), by definition we have

Lemma 2.8. The first obstruction fo(F) is the generator of the d-dimensional part of the obstruction ideal O_{F}^{d} = O_{F} ∩ H^{d}(BG). Moreover O_{F}^{i} = 0 for i < d. (Similar statement holds for fo(E) as well)

The last statement of the lemma comes in the following way: if O_{F}^{j} ≠ 0 for some j < d − 1 then E has no section, and it has no section over sk_{j}M either (because i_{j}^{∗} : H^{j}(M) → H^{j}(sk_{j}M) is injective ) which is a contradiction, since π_{j−1}(F) = 0, so every section can be extended over sk_{j}M.

Now let us suppose, that we have an open embedding of F into a contractible G-space V. (In most applications F is given as an open subset of a vector space.) Let s be a generic section, and

F^{c}(s) := {m ∈ M : s(m) ∉ F }

We cannot say, that F^{c}(s) represents a cohomology element in the usual way, because (for example) it is not smooth in general. So we have to extend the notion of Poincaré dual in the following way, see [?]

Definition 2.9. [?] Let X be a closed subset of a topological space Y. X represents the cohomology class [X] ∈ H^{d}(Y; ℤ) if for the inclusion i : Y \ X → Y

ker H^{j}(i) ≅ \begin{cases}0 & if j < d \\ ℤ & if j = d \end{cases}

Then [X] is the generator of ker H^{d}(i).

The class [X] is defined up to sign, we can call the two choices orientations. If f : X → Y is a proper map of smooth oriented manifolds, then Thom isomorphism for the normal bundle of X shows that X represents the ordinary Poincaré dual of X, so this is indeed a generalization. Moreover, if A ⊂ X is a complex algebraic subvariety, then A will always represent a dual class in X, and we will use this case of the definition in every case.

If we use this definition for X = P ×_{G} (V \ F) ⊂ P ×_{G} V and Y = P ×_{G} V then since H^{∗}(P ×_{G} V) = H^{∗}(M) we get that this generalized Poincaré dual is exactly the first obstruction: [P ×_{G} (V \ F)] = fo(E). Now by the naturality property of the Poincaré dual we have the following, see [?]

Theorem 2.10. Suppose that the Lie group G acts on the vector space V and F is a G-invariant open subspace of V. Assume moreover that π_{i}(F) = 0 for i < d − 1 and π_{d−1}(F) ≅ ℤ and that for the principal G-bundle P → M the cohomology class fo(P ×_{G} F) is not zero and F^{c}(s) represents a cohomology class in H^{d}(M; ℤ). Then fo(P ×_{G} F) = ±[F^{c}(s)].

2.4. The definition of Thom polynomials. Thom polynomials are special first obstructions. Given a representation ρ : G → GL(V) we are looking for obstructions of having a section of a V -bundle associated to this representation avoiding a certain orbit η (or more generally a G-invariant subset of V). In effect we want to avoid the closure of η. In our examples G will be a complex Lie group acting on the complex vector space V, and η ⊂ V is a G-invariant algebraic subvariety of V. Then the assumptions stand for V \ η, so it is highly connected by the following observation (see [?])

Observation 2.11. Let X ⊂ ℂ^{N} be a d (real) codimensional complex algebraic variety. Then

π_{i}(ℂ^{N} \ X) ≅ \begin{cases}0 & if i < d − 1 \\ ℤ & if i = d − 1. \end{cases}

(If d = 2 then π_{1} should be replaced by H_{1}.)

Remark 2.12. Similar theorem holds for real varieties with some extra condition. However the closure of an orbit of a real representation is not necessarily a real algebraic variety, the algebraic closure can contain some other orbits with the same codimension.

Now we can define Thom polynomials:

Definition 2.13. (Thom polynomial) For an invariant complex algebraic subvariety of the complex representation ρ : G → GL(V) we define

Tp(η) = fo(EG ×_{G} (V \ \overline{η})).

In other words Tp(η) is a generator of the d-dimensional part of the so called avoiding ideal, which is simply O_{V \\overline{η}}.

Now the assumptions of ?? hold, so we have

Lemma 2.14. Let η ⊂ V be an invariant complex algebraic subvariety of the complex representation ρ : G → GL(V), where G is a complex Lie group. Then

T p(η) = [\overline{B_{G}η}] ∈ H^{∗}(BV) = H^{∗}(BG).

Remark 2.15. Thom polynomials are generalizations of the Euler class, because by definition T p(0 ∈ V) = e_{G}(V), where e_{G}(V) denotes the Euler class of BV → BG.

3. Representations of Lie groups and Lie algebras

In this section we give a short summary of the representations of Lie groups and Lie algebras. We will focus on Lie algebras, which will use quite a lot later. In this section we follow more or less the structure of the work [?] We deal only with complex Lie groups and Lie algebras. A representation of a Lie group G on the complex vector space V is simply a homomorphism of Lie groups G → GL(V). A representation of a Lie algebra 𝔤 on a complex vector space V is a Lie algebra (i.e Lie-bracket preserving) map 𝔤 → 𝔤𝔩(V) = End(V). A fundamental theorem says, that representations of a connected, simply connected Lie group are in one-to one correspondence with representations of its Lie algebra: by associating ρ : G → GL(V) its differential dρ : 𝔤 → 𝔤𝔩(V). We can form tensor products, symmetric powers, exterior powers of representations in a natural way.

A Lie algebra is simple, if it contains no nontrivial ideals. The lower central series of subalgebras defined by 𝔇_{1}𝔤 = [𝔤, 𝔤], and 𝔇_{k}𝔤 = [𝔤, 𝔇_{k−1}𝔤], and the derived series by 𝔇^{1}𝔤 = [𝔤, 𝔤] and 𝔇^{k}𝔤 = [𝔇^{k−1}𝔤, 𝔇^{k−1}𝔤].

Definition 3.1. A Lie algebra is

(i) nilpotent, if 𝔇_{k}𝔤 = 0 for some k; (ii) solvable, if 𝔇^{k}𝔤 = 0 for some k; (iii) perfect, if 𝔇𝔤 = 𝔤; (iv) semisimple, if 𝔤 has no nonzero ideals;

(v) the radical Rad(𝔤) of 𝔤 is the maximal solvable ideal in 𝔤. We say that a Lie group has one of this properties, if its Lie algebra has.

The standard example for nilpotent and solvable Lie algebra is the algebra of strictly upper-triangular n×n matrices and Lie algebra of upper-triangular matrices, respectively. Every Lie algebra fits into an exact sequence

0 → Rad(𝔤) → 𝔤 → 𝔤/Rad(𝔤) → 0

where the first algebra is solvable, the last is semisimple. Now by Lie’s theorem an irreducible representation of a solvable Lie algebra is one dimensional, and the following consequence of the exact sequence says, that it is enough to know the representations of semisimple Lie algebras:

Theorem 3.2. If 𝔤 is a complex Lie algebra, then every irreducible representation of 𝔤 is of the form V = V_{0} ⊗ L where V_{0} is an irreducible representation of 𝔤/Rad(𝔤) and L is a one-dimensional representation of 𝔤.

Complete reducibility – that is, every representation decomposes into a sum of irreducible representations – is not true in general, but for semisimple Lie group it is true. One more note about this classification: we can describe all the simple Lie algebras. With five exceptions every simple complex Lie algebra is isomorphic to either _{n}ℂ, 𝔰𝔬_{n}ℂ, 𝔰𝔭_{2n}ℂ, these are the classical Lie algebras. The five exceptions are denoted by 𝔤_{2}, 𝔣_{4}, 𝔢_{6}, 𝔢_{7} and 𝔢_{8}, these are the exceptional Lie algebras.

For every Lie algebra 𝔤 there is a unique simply connected Lie group G̃ with this Lie algebra (called simply connected form) and all other complex Lie groups with Lie algebra 𝔤 are quotients of G by a discrete subgroup of the center Z(G). (G/Z(G) is called the adjoint form.) For any representation ρ : 𝔤 → 𝔤𝔩(V) corresponds a representation ρ̃ of G̃ by ρ̃(exp(X)) = exp(ρ(X)). For any other group G = G̃/C with C < Z(G) the representations of G are the representations of G̃ trivial on C. We will turn back to the representations of Lie groups, but we first start with Lie algebras.

3.1. Representations of Lie algebras. (reference:[?])

Let 𝔤 = LieG be a semisimple Lie algebra, and let 𝔥 ⊂ 𝔤 be its Cartan-subalgebra, that is, the maximal abelian subalgebra of 𝔤, then any action of 𝔥 on any representation of 𝔤 will be diagonalizable. Let 𝔥 act on 𝔤 by the adjoint representation, and decompose 𝔤 accordingly, this is the Cartan-decomposition

𝔤 = 𝔥 ⊕ (⊕𝔤_{α})

where the action of 𝔥 preserves each 𝔤_{α} and acts on it by scalar multiplication by the linear functional α ∈ 𝔥^{∗}, that is, for any H ∈ 𝔥 and X ∈ 𝔤_{α} we will have

ad(H)(X) = α(H)X

The second direct sum in the expression above is over a finite set of eigenvalues α ∈ 𝔥^{∗}, these are called the roots of the Lie-algebra, and the corresponding subspaces 𝔤_{α} are called the root spaces. 𝔥 is the eigenspace to the eigenvalue 0. R will denote the set of roots. The adjoint action of 𝔤_{α} carries the eigenspace 𝔤_{β} into another eigenspace 𝔤_{α+β}.

Proposition 3.3. (i) Each root space 𝔤_{α} is one dimensional (ii) R generates a lattice Λ_{R} ⊂ 𝔥^{∗} of rank equal to the dimension of 𝔥. (iii) R is symmetric about the origin, i.e., if α is a root then −α is a root as well.

Let V be an irreducible representation of 𝔤, then V has a direct sum decomposition

V = ⊕V_{α}

where the direct sum runs over a finite set of α ∈ 𝔥^{∗} and 𝔥 acts diagonally on each V_{α} by multiplication by the eigenvalue α, so for H ∈ 𝔥 and v ∈ V_{α}

H(v) = α(H)v.

The eigenvalues in the direct sum decomposition are called the weights of V, and V_{α} are the weight spaces, the dimension of V_{α} is the multiplicity of α. The action of the rest of 𝔤 on V can be described in these terms: for any root β

𝔤_{β} : V_{α} → V_{α+β}

so we can think of the action of 𝔤_{β} on V as a translation in the weight diagram. The weights of an irreducible representation are congruent modulo the the root lattice Λ_{R}.

Now we can find distinguished subalgebras s_{α} ≅ 𝔰𝔩_{2}ℂ ⊂ 𝔤 in the following way: Let 𝔤_{α} ⊂ 𝔤 be a root space, then by proposition ?? there is another root space 𝔤_{−α} ⊂ 𝔤 and [𝔤_{α}, 𝔤_{−α}] is a subspace of 𝔥 of dimension one. Now the subalgebra

𝔰_{α} = 𝔤_{α} ⊕ 𝔤_{−α} ⊕ [𝔤_{α}, 𝔤_{−α}]

is isomorphic to 𝔰𝔩_{2}(ℂ). So we can pick a basis X_{α} ∈ 𝔤_{α}, Y_{α} ∈ 𝔤_{−α} and H_{α} ∈ [𝔤_{α}, 𝔤_{−α}], satisfying the standard commutation relations

[H_{α}, X_{α}] = 2X_{α}, [H_{α}, Y_{α}] = −2Y_{α}, [X_{α}, Y_{α}] = H_{α}

and H_{α} is uniquely determined by these relations. These H_{α} ∈ 𝔥 elements are called the distinguished elements. All eigenvalues of the action of H_{α} must be integers, so if Λ_{W} denotes the following lattice

Λ_{W} = {β ∈ 𝔥^{∗} : β(H_{α}) ∈ ℤ for all α}

called the weight lattice then

Proposition 3.4. All weights of all representations of 𝔤 will lie in Λ_{W}.

Then Λ_{R} ⊂ Λ_{W} is a sub-lattice of finite index. We have the following for the weights of tensor products, exterior and symmetric powers and duals of representations:

Proposition 3.5. Let V, W be two representations of 𝔤, the weights of V and W are v_{1}, …, v_{n}, and w_{1}, …, w_{m} respectively. Then we have

(i) The weights of V ⊗ W are {v_{i} + w_{j} : 1 ≤ i ≤ n, 1 ≤ j ≤ m}. (ii) The weights of ∧^{k}V are {v_{i_{1}} + v_{i_{2}} + … + v_{i_{k}} : 1 ≤ i_{1} < … < i_{k} ≤ n}. (iii) The weights of Sym^{k}V are {v_{i_{1}} + v_{i_{2}} + … + v_{i_{k}} : 1 ≤ i_{1} ≤ … ≤ i_{k} ≤ n}. (iv) The weights of V^{∗}, the dual representation of V are {−v_{i} : 1 ≤ i ≤ n}.

For every α we can introduce an involution W_{α} of 𝔥^{∗}, which is the reflection in the hyperplane Ω_{α} = {β ∈ 𝔥^{∗} : β(H_{α}) = 0}, that is

W_{α}(β) = β − \frac{2β(H_{α})}{α(H_{α})}α = β − β(H_{α})α.

The group generated by these involutions is called the Weyl-group of the Lie-algebra.

Proposition 3.6.

(i) The set of weights of any representation of 𝔤 is invariant under the Weyl-group. (ii) Every element of the Weyl-group is induced by an automorphism of the Lie-

algebra 𝔤 carrying 𝔥 to itself.

Now we can define the Killing form, which is a symmetric bilinear form on 𝔤, it will be denoted by B. For X, Y ∈ 𝔤

B(X, Y) = T race(ad(X) ∘ ad(Y) : 𝔤 → 𝔤)

Proposition 3.7.

(i) B is positive definite on the real subspace of 𝔥 spanned by the vectors H_{α} : α ∈ R. (ii) The Killing form is nondegenerate on the semisimple Lie algebra 𝔤.

The Killing form on 𝔥 defines a symmetric bilinear form on 𝔥^{∗}. The element T_{α} of 𝔥 which corresponds to α ∈ 𝔥^{∗} by the Killing form satisfies by definition the condition

B(T_{α}, H) = α(H) for all H ∈ 𝔥

It can be seen easily, that T_{α} = (2/B(H_{α}, H_{α})H_{α}, and so the Killing form on 𝔥^{∗} is defined by B(α, β) = B(T_{α}, T_{β})

The dual of the previous proposition is the following.

Proposition 3.8. The Killing form B is positive definite on the real vector space spanned by the root lattice Γ_{R}.

Proposition 3.9. The space 𝔥^{∗} is an irreducible representation of the Weyl group.

We can characterize the Weyl group as up to scalars it is the unique inner product on 𝔥^{∗} preserved by the Weyl-group, i.e, in terms of which the the Weyl group acts as a group of orthogonal transformations. The Weyl-group is the group generated by the reflections in the hyperplanes perpendicular to the roots of the Lie algebra with respect to the Killing form.

Now choose a real linear functional l on the lattice Λ_{R} irrational with respect to this lattice. This gives us a decomposition of the set

R = R^{+} ∪ R^{−}

where R^{+} = {α : l(α) > 0}, these are called positive roots, the remaining are the negative roots, this decomposition is called an ordering of the roots.

Definition 3.10. Let V be a representation of 𝔤. A nonzero vector v ∈ V that is both an eigenvector for the action of 𝔥 and in the kernel of 𝔤_{α} for all α ∈ R^{+} is called a highest weight vector of V.

Proposition 3.11. For any semisimple complex Lie algebra 𝔤

(i) Every finite dimensional representation V of 𝔤 possesses a highest weight vector.

(ii) The subspace W of V generated by the images of a highest weight vector v under successive applications of root spaces 𝔤_{β} for β ∈ R^{−} is an irreducible subrepresentation.

(iii) An irreducible representation possesses a unique highest weight vector up to scalars.This is called the highest weight of the irreducible representation.

We say that a positive (resp. negative) root α primitive or simple if it cannot be expressed as a sum of two positive (resp. negative) roots.

Proposition 3.12.

(i) Any irreducible representation V is generated by the images of its highest weight vector v under successive applications of root spaces 𝔤_{β} where β ranges over the primitive negative roots.

(ii) The set of weights of V will be exactly the weights that are congruent to α modulo the root lattice Λ_{R} and that lie in the convex hull of the images of α under the Weyl group.

The highest weight of any representation of V will be a weight α satisfying α(H_{γ}) ≥ 0 for every γ ∈ R^{+}, that is in terms of the Killing form, it makes an acute or right angle with each of the positive roots. The locus W in the real span of the roots of these points is called the Weyl chamber associated to the ordering of the roots. The Weyl group acts transitively on the set of Weyl chambers. Now we can classify all the irreducible representations of 𝔤:

Theorem 3.13. For any α in the intersection of the Weyl chamber W with the weight lattice Λ_{W} there exist a unique irreducible, finite dimensional representation Γ_{α} of 𝔤 with highest weight α; this gives a bijection between W ∩ Λ_{W} and the set of irreducible representations of 𝔤. The weights of Γ_{α} will consist of those elements of the weight lattice congruent to α modulo the root lattice Λ_{R} and lying in the convex hull of the set of points in 𝔥^{∗} conjugate to α under the Weyl group.

There are fundamental weights ω_{1}, …, ω_{n} such that any dominant weight can be expressed uniquely as a nonnegative integral linear combination of them. They can be characterized geometrically as the first weights met along the edges of the Weyl chamber, or algebraically as those elements ω_{i} in 𝔥^{∗} such that ω_{i}(H_{α_{j}}) = δ_{i,j} where α_{1}, …, α_{n} are the simple roots.

3.2. Representation of Lie groups. Remember, that the representation of a semisimple Lie group G = G̃/C where C < Z(G) is simply a representation of G̃ trivial on C. The center C lies in the image of the Cartan subalgebra under the exponential map. This image, which is the maximal abelian subgroup of G, called Cartan subgroup of G, or maximal torus denoted by H. Then 𝔥 = LieH. Now we can define the dual lattices of Λ_{W} and Λ_{R}, (the weight lattice and root lattice) namely:

Γ_{R} = ℂ{H_{γ} : γ ∈ R} ⊂ 𝔥, Γ_{W} = {X ∈ 𝔥 : α(X) ∈ ℤ for all α ∈ R} ⊂ 𝔥

There is a natural pairing from Γ_{W} /Γ_{R} to the center of G̃, which is given by the exponential. More precisely, let e_{G} : 𝔥 → H ⊂ G be the homomorphism defined by e_{G}(X) = exp(2πiX). When G = G̃ is the simply-connected group, then Ker(eG̃) = Γ_{R} and eG̃(Γ_{W}) is the center of G̃, and eG̃ induces an isomorphism Γ_{W} /Γ_{R} ≅ Z(G̃). More generally, for any G = G̃/C, define a lattice

Γ(G) = Ker(e_{G}) ⊂ 𝔥,

then e_{G} determines an isomorphism

Γ_{W} /Γ(G) ≅ Z(G),

and we have Theorem 3.14. There is a one-to one correspondence between connected Lie groups G with Lie algebra 𝔤 and lattices Λ(G) ⊂ 𝔥^{∗} such that Λ_{R} ⊂ Λ(G) ⊂ Λ_{W}. The correspondence is given by associating to a group G the lattice dual to the kernel of the exponential map. In terms of this correspondence, the irreducible representation V_{λ} of 𝔤 with highest weight λ ∈ 𝔥^{∗} will lift to a representation of the group G corresponding to Λ(G) ⊂ 𝔥^{∗} iff λ ∈ Λ(G).

Note also, that

H ≅ 𝔥/Γ(G) ≅ ℂ^{∗} × ⋯ ℂ^{∗}

with n = dim_{ℂ}𝔥 copies of ℂ^{∗}.

Now elements of the lattice Λ(G) ⊂ Λ_{W} are in bijection with the set of characters of H. (a character is a Lie-group homomorphism H → U(1). The bijection is given by w ∈ Λ(G) → exp(w) ∈ U(1), and conversely, the differential of a homomorphism H → U(1) is a weight w ∈ Λ(G) : 𝔥 → ℝ. Given a representation ρ : G → GL(V) and T ⊂ G is a maximal torus, 𝔱 = LieT, then we have restriction ρ|T : T → GL(V), and V decomposes into the direct sum of one-dimensional eigenspaces, and this decomposition is the same as for the action of the Lie algebra 𝔱, in the weight spaces V_{α} T acts by multiplication by exp(2πiα) ∈ U(1), thus we can associate a set of weights W = {w_{1}, …, w_{n}} ⊂ Λ(G) to any representation ρ of G. The Weyl group 𝔚, which we defined as a subgroup of Aut(𝔥^{∗}) can be described for the Lie group G in the following way: let N(H) be the normalizer of the Cartan subgroup, which is the greatest subgroup of G in which H is a normal subgroup. Then we have homomorphisms:

N(H) → Aut(H) → Aut(𝔥) → Aut(𝔥^{∗})

the first defined by conjugation, the second by differentiation at the identity, and the third using the identification of 𝔥 and 𝔥^{∗} via the Killing form. This map determines an isomorphism N(H)/N ≅ 𝔚.

3.3. Calculation of equivariant cohomologies. Here we present some methods of calculating equivariant cohomologies. First of all a theorem of Borel tell us how to compute the equivariant cohomology of the point. We work with coefficients ℚ or ℝ. Let T ⊂ G be a maximal torus. As we have seen, there is a bijection between the weight lattice Γ_{W} , and one dimensional irreducible representations, namely, if λ ∈ Γ_{W} then the associated representation is E_{λ} = ET ×_{ρ_{λ}} ℂ^{1} → BT. (the action ρ_{λ} : T → Aut(ℂ^{1}) is multiplication by exp(2πiλ)). So we have a map

f_{G} : Γ_{W} → H^{2}(BT)
λ → c_{1}(E_{λ}).

This is a homomorphism: if λ, μ ∈ Γ_{W} then ρ_{λ+μ} = ρ_{λ}ρ_{μ}, and E_{ρ_{λ}ρ_{μ}} = E_{ρ_{λ}} ⊗ E_{ρ_{μ}}, whose first Chern class is the sum of the Chern classes of the components. So f_{G}(λ + μ) = c_{1}(E_{λ}) + c_{1}(E_{μ}), so f_{G} is indeed a homomorphism. When V is a representation of G, and the weights of this representation are {w_{1}, …, w_{m}} (which are the Chern classes of the one-dimensional representations of the action of the maximal torus of G), then the Chern classes of the bundle BV → BG—which are called equivariant Chern classes—are

c_{i}^{G}(V) = σ_{i}(w_{1}, …, w_{m})

where σ_{i} is the ith elementary symmetric polynomial. An important special case of this the equivariant Euler class e_{G}(V) = w_{1} ⋯ w_{m}. If T^{n} = U(1)×U(1)×…×U(1) is a complex torus, then we have H_{G}^{T}pt = H^{∗}(BT) = H^{∗}(BU(1) × … × BU(1)) = ℝ[α_{1}, …, α_{n}] where α_{i} correspond to the first Chern class of the principal complex line bundle ℂ^{1} → γ_{ℂ}^{1} → BU(1), these are the generators of the cohomology ring. For arbitrary G Lie group we have

Theorem 3.15. (Borel) Let T ⊂ G be a maximal torus of the complex Lie group G, which is a maximal compact abelian subgroup. Then the map

i^{∗} : H^{∗}(BG, ℚ) → H^{∗}(BT, ℚ)

is injective, and Im i^{∗} = [H^{∗}(BT)]^{W}, so

H^{∗}(BG) = [H^{∗}(BT)]^{𝔚} = ℝ[α_{1}, …, α_{n}]^{𝔚}

where the upper index means the 𝔚-invariant part of the ring. (The Weyl group acts on the weight lattice, and α_{i} are elements of the weight lattice.)

4. Representations of the classical Lie algebras

The standard reference for this section is [?].

4.1. Analysis of 𝔰𝔩_{n}ℂ. As we have seen, 𝔰𝔩_{n} is the subalgebra of n × n traceless matrices. The Cartan subalgebra is the subalgebra of diagonal matrices. Let E_{i,i} denote the diagonal matrix with all 0-s in the diagonal except for the ith entry, which is 1, so that takes e_{i} to itself and kills e_{j} for i ≠ j. Then

𝔥 = {a_{1}H_{1} + a_{2}H_{2} + … + a_{n}H_{n} : a_{1} + … + a_{n} = 0}
𝔥^{∗} = ℂ{L_{1}, L_{2}, …, L_{n}}/(L_{1} + … + L_{n} = 0)

where L_{i}(H_{j}) = δ_{i,j}, and so L_{i} ∈ 𝔥^{∗}. (The notation ℂ{} means ℂ⟨⟩, so the generated space). If E_{i,j} is the endomorphism of ℂ^{n} carrying e_{j} to e_{i} and killing e_{k} for all k ≠ j, then

ad(a_{1}H_{1} + a_{2}H_{2} + … + a_{n}H_{n}) = (a_{i} − a_{j})E_{i,j}

so E_{i,j} is an eigenvector for the action of 𝔥 with eigenvalue L_{i} − L_{j} ∈ 𝔥^{∗}: the roots of 𝔰𝔩_{n}ℂ are just the pairwise differences of the L_{i}: R = {L_{i} − L_{j} : 1 ≤ i, j ≤ n}, and the root spaces are ℂE_{i,j}. The root lattice Λ_{R} ⊂ 𝔥^{∗} can thus be described as

Λ_{R} = {∑ a_{i}L_{i} : a_{i} ∈ ℤ, ∑ a_{i} = 0}/(∑ L_{i} = 0).

We can now calculate the Killing form directly: by the definition we have B(X, Y) = ∑_{α∈R} α(X)α(Y) so

B(∑ a_{i}H_{i}, ∑ b_{i}H_{i}) = ∑_{i≠j}(a_{i} − a_{j})(b_{i} − b_{j}) = ∑_{i} ∑_{j≠i}(a_{i}b_{i} + a_{j}b_{j} − a_{i}b_{j} − a_{j}b_{i})

Since ∑_{i≠j} a_{j} = −a_{i} and ∑_{i≠j} b_{j} = −b_{i} this simplifies to

B(∑ a_{i}H_{i}, ∑ b_{i}H_{i}) = 2n ∑ a_{i}b_{i}.

So for the dual form on 𝔥^{∗}:

B(∑ a_{i}L_{i}, ∑ b_{i}L_{i}) = (1/2n)(∑_{i} a_{i}b_{i} − (1/n) ∑_{i,j} a_{i}b_{j}).

The Weyl group in this case is the symmetric group σ_{n}, and this Killing form is the unique one up to scalars invariant under this group. The L_{i} all have the same length, and the angles between all pairs are the same. To picture the roots in 𝔥^{∗} we can think of the points L_{i} situated at the vertices of a regular (n − 1)-simplex with the origin located at the barycenter of that simplex.The next step, finding the distinguished subalgebras 𝔰_{α}. The root space 𝔤_{L_{i}−L_{j}} corresponding to the root L_{i} − L_{j} is generated by E_{i,j}, so the subalgebra 𝔰_{L_{i}−L_{j}} is generated by

E_{i,j}, E_{j,i} and [E_{i,j}, E_{j,i}] = H_{i} − H_{j}.

The eigenvalue of H_{i} − H_{j} acting on E_{i,j} is (L_{i} − L_{j})(H_{i} − H_{j}) = 2, so that the corresponding distinguished element H_{L_{i}−L_{j}} in 𝔥 is H_{i} − H_{j}. The annihilator is the hyperplane Ω_{L_{i}−L_{j}} = {∑ a_{i}L_{j} : a_{i} = a_{j}}. Knowing the H_{α} we can determine the weight lattice: the linear functional ∑ a_{i}L_{i} ∈ 𝔥^{∗} to have integral values on all the distinguished elements, it is necessary and sufficient that all the a_{i} be congruent to one another modulo ℤ. Since ∑ L_{i} = 0 the weight lattice is the following:

Λ_{W} = ℤ{L_{1}, …, L_{n}}/(∑ L_{i} = 0).

So the weight lattice of 𝔰𝔩_{n}ℂ may be realized as the lattice generated by the vertices of a regular (n − 1)-simplex Δ centered at the origin, and the roots as the pairwise differences of these vertices.

Remark 4.1. Since the lattice Λ_{W} can be generated by Λ_{R} together with any of the vertices L_{i} of our simplex, Λ_{W} /Λ_{R} is cyclic, generated by an L_{i}, and it can be shown, that Λ_{W} /Λ_{R} = ℤ/nℤ.

We can see, that the reflection in the hyperplane perpendicular to the root L_{i} −L_{j} exchanges L_{i} and L_{j}, and fixes the other L_{k}, so this is a transposition on {L_{i}}, and the Weyl group is the symmetric group acting on the generators L_{i} of 𝔥^{∗}. Let us choose a direction, that is, a linear functional l on 𝔥^{∗} which has the following form:

l(∑ a_{i}L_{i}) = ∑ c_{i}a_{i}

with ∑ c_{i} = 0, and c_{1} > … > c_{n}. This determines an ordering of 𝔥^{∗}, and of the roots with

R^{+} = {L_{i} − L_{j} : i < j}, R^{−} = {L_{i} − L_{j} : j < i}.

The primitive negative roots are the roots L_{i+1} −L_{i} (i = 1, ...n−1), and the (closed) Weyl chamber associated to this ordering is

𝔚 = {∑ a_{i}L_{i} : a_{1} ≥ a_{2} ≥ … ≥ a_{n}}.

Geometrically, if we take the barycentric subdivision of the faces of the simplex Δ, the Weyl chamber will be the cone over one (n − 2)-simplex of the barycentric subdivision. The edges of the Weyl chamber are the lines generated by the vectors

L_{1}, L_{1} + L_{2}, L_{1} + L_{2} + L_{3}, …, L_{1} + … + L_{n−1} = −L_{n},

these are the fundamental weights. The faces of 𝔚 will thus be the hyperplanes

Ω_{L_{i}−L_{i+1}} = {∑ a_{j}L_{j} : a_{i} = a_{i+1}}

perpendicular to the primitive negative roots L_{i+1}−L_{i}. We can index the irreducible representations of 𝔰𝔩_{n}ℂ in the following way: Γ_{a_{1},... ,a_{n−1}} with a_{1}, …, a_{n−1} ∈ ℕ denotes the irreducible representation with highest weight a_{1}L_{1} + a_{2}(L_{1} + L_{2}) + … + a_{n−1}(L_{1} + … + L_{n−1}). As a consequence of Proposition ?? Γ_{a_{1},... ,a_{n−1}} will occur as a subrepresentation of the tensor product of symmetric powers

Sym^{a_{1}}V^{(1)} ⊗ Sym^{a_{2}}V^{(2)} ⊗ ⋯ ⊗ Sym^{a_{n−1}}V^{(n−1)}

where V^{(i)} = Γ_{0,... ,1,...0} denotes the irreducible representation corresponding to the fundamental weight ω_{i} = L_{1}+…+L_{i}. Now it can be proved, that ∧^{k}V is irreducible, where V = ℂ^{n} is the standard representation of 𝔰𝔩_{n}ℂ, so V^{(i)} = Γ_{0,... ,1,...0} = ∧^{k}V.

Proposition 4.2. The exterior powers ∧^{k}V of the standard representations V are irreducible for k = 1, …, n, and these are the fundamental representations Γ_{0,... ,0,1,0,... ,0} = Γ_{ω_{i}} = ∧^{k}V.

We will use moreover the symmetric powers of the standard representations, which are irreducible representations as well: Γ_{k,0,... ,0} = Sym^{k}V.

4.2. Analysis of 𝔰𝔭_{n}ℂ. Let V be a 2n-dimensional complex vector space, Q a nondegenerate, skew-symmetric bilinear form on V. Then as we have seen

Sp_{2n}ℂ = {A ∈ Aut V : Q(Av, Aw) = Q(v, w), v, w ∈ V },
𝔰𝔭_{n}^{ℂ} = {A ∈ End V : Q(Av, w) + Q(v, Aw) = 0, v, w ∈ V }

Let e_{1}, …, e_{2n} be a basis such that Q(e_{i}, e_{i+n}) = 1, Q(e_{i+n}, e_{i}) = −1, Q(e_{i}, e_{j}) = 0 if j ≠ i ± j, so the matrix of Q in this basis is M = ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ). Then

Sp_{2n}ℂ = {A ∈ GL_{2n}ℂ : M = A^{t}MA}, 𝔰𝔭_{2n}ℂ = {A ∈ Mat_{2n}ℂ : A^{t}M + MA = 0},

and we get that

𝔰𝔭_{2n}ℂ = ( \begin{matrix}A & B \\ C & D \end{matrix} )

with B, C symmetric, A = −D^{t}. The Cartan subalgebra is again the subalgebra of diagonal matrices, so 𝔥 = ℂ{H_{i} : 1 ≤ i ≤ n} where H_{i} = E_{i,i} − E_{n+i,n+i}, and the dual basis elements of 𝔥^{∗} are L_{i}, (i = 1, …, n) with ⟨L_{i}, H_{j}⟩ = δ_{i,j}. Now it can be easily checked that:

Proposition 4.3. X_{i,j} = E_{i,j} − E_{n+j,n+i} ∈ 𝔰𝔭_{2n}ℂ is an eigenvector for the action of 𝔥 with eigenvalue L_{i} − L_{j}. Y_{i,j} = E_{i,n+j} + E_{j,n+i} is an eigenvector for the action of 𝔥 with eigenvalue L_{i} + L_{j}. Z_{i,j} = E_{n+i,j} +E_{n+j,i} is an eigenvector for the action of 𝔥 with eigenvalue −L_{i} −L_{j}. U_{i} = E_{i,n+i} and V_{i} = E_{n+i,i} are eigenvectors for the action of 𝔥 with eigenvalue 2L_{i} and −2L_{i}, respectively.

So the roots are R = {±L_{i} ± L_{j}}. In similar way, as in the case of 𝔰𝔩_{n}ℂ we can compute the Killing form on 𝔥: for H = ∑ a_{i}H_{i} and H^{∗} = ∑ b_{i}H_{i} we have

B(H, H^{∗}) = (4n + 4)(∑ a_{i}b_{i})

and the corresponding form on 𝔥^{∗} is

B(∑ a_{i}L_{i}, ∑ b_{i}L_{i}) = (1/(n + 1)) ∑ a_{i}b_{i}.

It can be easily shown, that for the roots L_{i} − L_{j} and L_{j} − L_{i} corresponding to the elements X_{i,j} and X_{j,i} we have [X_{i,j}, X_{j,i}] = H_{i} − H_{j}, so the distinguished element H_{L_{i}−L_{j}} is a multiple of H_{i}−H_{j} and ad(H_{i}−H_{j})(X_{i,j}) = 2X_{i,j}, so H_{L_{i}−L_{j}} = H_{i}−H_{j}. In a similar way we get:

Proposition 4.4. The distinguished elements of 𝔰𝔭_{2n}ℂ are

H_{L_{i}−L_{j}} = H_{i} − H_{j}, H_{L_{i}+L_{j}} = H_{i} + H_{j},
H_{−L_{i}−L_{j}} = −H_{i} − H_{j}, H_{2L_{i}} = H_{i}, H_{−2L_{i}} = −H_{i}.

As a corollary we have the weight lattice:

Λ_{W} = {∑ a_{i}L_{i} : a_{i} ∈ ℤ}

Now the index Λ_{W} : Λ_{R} is 2. The Weyl group action on 𝔥^{∗} is generated by reflections: the reflection in the plane Ω_{2L_{i}} given by ⟨H_{i}, L⟩ = 0 reverse the sign of L_{i} and leaves the other L_{j} fixed; reflection in the plane ⟨H_{i} − H_{j}, L⟩ = 0 exchange L_{i} and L_{j} and leave the remaining L_{k} alone, so the action can be characterized by a permutation of the L_{i}-s, and changing the signs some of them, so it fits into the sequence:

1 → (ℤ_{2})^{n} → 𝔚 → σ_{n} → 1.

and this sequence splits: 𝔚 is the semi direct product of σ_{n} and ℤ_{2}^{n}. Choosing an ordering l(∑ a_{i}L_{i}) = ∑ c_{i}a_{i}, c_{1} > … > c_{n} > 0, the positive roots are

R^{+} = {L_{i} + L_{j}}_{i≤j} ∪ {L_{i} − L_{j}}_{i<j}

with primitive positive roots {L_{i} − L_{i+1}}_{i=1,... ,n−1} and 2L_{n}. The corresponding closed Weyl chamber is

W = {a_{1}L_{1} + … + a_{n}L_{n} : a_{1} ≥ … ≥ a_{n} ≥ 0}.

The walls of this chamber- the maximal dimensional faces are

{∑ a_{i}L_{i} : a_{1} > … > a_{i} = a_{i+1} > … > a_{n} > 0}, {∑ a_{i}L_{i} : a_{1} > … > a_{n} = 0}

and the edges are

{∑ a_{i}L_{i} : a_{1} = … = a_{i} > a_{i+1} = … = a_{n} = 0}.

The fundamental weights are ω_{i} = L_{1} + … + L_{i}, and as before we denote the irreducible representation with highest weight a_{1}ω_{1} + … + a_{n}ω_{n} = (a_{1} + … + a_{n})L_{1} + (a_{2} + ⋯ + a_{n})L_{2} + … + a_{n}L_{n} with Γ_{a_{1},... ,a_{n}}. In the simplectic case ∧^{k}V is not an irreducible representation. We have the natural contraction map

φ_{k} : ∧^{k}V → ∧^{k−2}V

defined by

φ_{k}(v_{1} ∧ … ∧ v_{k}) = ∑_{i<j} (v_{i}, v_{j})(−1)^{i+j−1}v_{1} ∧ … ∧ v̂_{i} ∧ … ∧ v̂_{j} ∧ … ∧ v_{k}

Since the representation ∧^{k−2}V does not have the weight L_{1}+…+L_{k}, the irreducible representation with this highest weight will have to be contained in the kernel of this map.

Theorem 4.5. For 1 ≤ k ≤ n, the kernel of the map φ_{k} is exactly the irreducible representation V^{(k)} = Γ_{0,... ,0,1,0,... ,0} with highest weight L_{1} + … + L_{k}.

Now every irreducible representation occurs in a tensor product of symmetric powers of these representations.

4.3. Analysis of 𝔰𝔬_{m}ℂ. In the case of orthogonal Lie algebras there is a difference between the so called even orthogonal Lie algebras 𝔰𝔬_{2n}ℂ and odd orthogonal Lie algebras 𝔰𝔬_{2n+1}ℂ.

Let V be a m-dimensional complex vector space, Q a nondegenerate, symmetric bilinear form on V. Then as we have seen

SO_{m}ℂ = {A ∈ GL(V) : Q(Av, Aw) = Q(v, w) v, w ∈ V, detA = 1},
𝔰𝔬_{n}^{ℂ} = {A ∈ End V : Q(Av, w) + Q(v, Aw) = 0 v, w ∈ V }

In case m = 2n let e_{1}, …, e_{2n} be a basis such that Q(e_{i}, e_{i+n}) = Q(e_{i+n}, e_{i}) = 1, Q(e_{i}, e_{j}) = 0 if j ≠ i ± n, so the matrix of Q in this basis is M = ( \begin{matrix}0 & I_{n} \\ I_{n} & 0 \end{matrix} ). Then

SO_{2n}ℂ = {A ∈ GL_{2n}ℂ : M = A^{t}MA}, 𝔰𝔬_{2n}ℂ = {A ∈ Mat_{2n}ℂ : A^{t}M + MA = 0},

and we get that

𝔰𝔬_{2n}ℂ = ( \begin{matrix}A & B \\ C & D \end{matrix} )

with B, C skew-symmetric, and A, D negative transposes of each other. When m = 2n + 1 we choose a basis e_{1}, …, e_{2n+1} in which the bilinear form has matrix M = ( \begin{matrix}0 & I_{n} & 0 \\ I_{n} & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ), and then

𝔰𝔬_{2n+1}ℂ = ( \begin{matrix}A & B & E \\ C & D & F \\ G & H & J \end{matrix} )

with B, C skew-symmetric, and A = −D^{t}, E = −H^{t}, F = −G^{t}, J = 0. The Cartan subalgebra is generated by the 2n × 2n diagonal matrices H_{i} = E_{i,i} − E_{n+i,n+i} (this is the same in the odd and the even case), the dual basis of 𝔥^{∗} is denoted by L_{i} where ⟨L_{j}, H_{i}⟩ = δ_{i,j}. Description of the roots:

Proposition 4.6. m=2n: X_{i,j} = E_{i,j} − E_{n+j,n+i} ∈ 𝔰𝔬_{2n}ℂ is an eigenvector for the action of 𝔥 with eigenvalue L_{i} − L_{j}. Y_{i,j} = E_{i,n+j} − E_{j,n+i} is an eigenvector for the action of 𝔥 with eigenvalue L_{i} + L_{j}. Z_{i,j} = E_{n+i,j} −E_{n+j,i} is an eigenvector for the action of 𝔥 with eigenvalue −L_{i} −L_{j}. So the roots of 𝔰𝔬_{2n}ℂ are {±L_{i} ± L_{j}}_{i≠j} ⊂ 𝔥^{∗}.

Proposition 4.7. m=2n+1: X_{i,j} = E_{i,j} − E_{n+j,n+i} viewed as an element of the Lie algebra 𝔰𝔬_{2n+1}ℂ is an eigenvector for the action of 𝔥 with eigenvalue L_{i} − L_{j}. Y_{i,j} = E_{i,n+j} − E_{j,n+i} viewed as an element of 𝔰𝔬_{2n+1}ℂ is an eigenvector for the action of 𝔥 with eigenvalue L_{i} + L_{j}. Z_{i,j} = E_{n+i,j} − E_{n+j,i} viewed as an element of 𝔰𝔬_{2n+1}ℂ is an eigenvector for the action of 𝔥 with eigenvalue −L_{i} − L_{j}.

U_{i} = E_{i,2n+1}−E_{2n+1,n+i} and V_{i} = E_{n+i,2n+1}−E_{2n+1,i} are eigenvectors for the action of 𝔥 with eigenvalue L_{i} and −L_{i}, respectively. So the roots of 𝔰𝔬_{2n}ℂ are {±L_{i} ± L_{j}}_{i≠j} ∪ {±L_{i}} ⊂ 𝔥^{∗}.

So the root diagram of 𝔰𝔬_{2n+1}ℂ is the same that of 𝔰𝔭_{2n}ℂ with the roots ±2L_{i} replaced by ±L_{i}, hence the Weyl group is the same as that of 𝔰𝔭_{2n}ℂ:

1 → (ℤ_{2})^{n} → 𝔚_{𝔰𝔬_{2n+1}ℂ} → σ_{n} → 1.

In the case of 𝔰𝔬_{2n}ℂ the Weyl group is the subgroup of 𝔚_{𝔰𝔭_{2n}ℂ} generated by reflection in the hyperplanes perpendicular to the roots ±L_{i} ± L_{j}, without the additional generator given by reflection in the roots ±L_{i}. So the kernel of this action, instead of acting as ±I on each of the coordinate axes independently, will consist of transformations of determinant 1, i.e, will act as −1 on an even number of axes. And every such transformation is indeed in the Weyl group, because reflection in Ω_{L_{i}+L_{j}} followed by reflection in Ω_{L_{i}−L_{j}} will send L_{i} to −L_{i}, L_{j} to −L_{j} and fix the other L_{k}. So the Weyl group of 𝔰𝔬_{2n}ℂ consist of transformations whose determinant agrees with the sign of the induced permutation of coordinate axes, and so we have the following exact sequence:

1 → (ℤ_{2})^{n−1} → 𝔚_{𝔰𝔬_{2n}ℂ} → σ_{n} → 1.

Choosing an ordering l(∑ a_{i}L_{i}) = ∑ c_{i}a_{i}, c_{1} > … > c_{n} > 0, the positive roots in the case of 𝔰𝔬_{2n+1}ℂ are

R^{+} = {L_{i} + L_{j}}_{i<j} ∪ {L_{i} − L_{j}}_{i<j} ∪ {L_{i}}_{i}

whereas in the case of 𝔰𝔬_{2n}ℂ

R^{+} = {L_{i} + L_{j}}_{i<j} ∪ {L_{i} − L_{j}}_{i<j}.

The primitive positive roots are

{{L_{i} − L_{i+1}}_{i=1,... ,n−1}, L_{n}} for 𝔰𝔬_{2n+1}ℂ
{{L_{i} − L_{i+1}}_{i=1,... ,n−1}, L_{n−1} + L_{n}} for 𝔰𝔬_{2n}ℂ.

In the first case the Weyl chamber is exactly the same as for 𝔰𝔭_{2n}ℂ, namely with m = 2n + 1

W = {∑ a_{i}L_{i} : a_{1} ≥ … ≥ a_{n} ≥ 0}.

In the case of 𝔰𝔬_{2n}ℂ since there is no root along the line spanned by the L_{i}, the equality a_{n} = 0 does not describe a face of the Weyl chamber, however, since L_{n−1} + L_{n} is still a root, we still have the inequality a_{n−1} + a_{n} ≥ 0, so

W = {∑ a_{i}L_{i} : a_{1} ≥ … ≥ a_{n−1} ≥ |a_{n}|}.

The Killing form on 𝔥 with the same computations

B(∑ a_{i}H_{i}, ∑ b_{i}H_{i}) = \{ \begin{matrix}(4n − 2) ∑ a_{i}b_{i} & if & m = 2n + 1 \\ (4n − 4) ∑ a_{i}b_{i} & if & m = 2n \end{matrix}

The corresponding Killing form on 𝔥^{∗} is

B(∑ a_{i}L_{i}, ∑ b_{i}L_{i}) = \{ \begin{matrix}4/(4n − 2) ∑ a_{i}b_{i} & if & m = 2n + 1 \\ 1/(n − 1) ∑ a_{i}b_{i} & if & m = 2n \end{matrix}

The distinguished elements H_{α} (α ∈ R) are the following:

Proposition 4.8. The distinguished elements of 𝔰𝔬_{m}ℂ are

H_{L_{i}−L_{j}} = H_{i} − H_{j}, H_{L_{i}+L_{j}} = H_{i} + H_{j},
H_{−L_{i}−L_{j}} = −H_{i} − H_{j},

and in the even case m = 2n

H_{L_{i}} = 2H_{i}, H_{−L_{i}} = −2H_{i}.

The weight lattice Λ_{W} is now for both even and odd orthogonal Lie algebras is the lattice generated by the L_{i} together with the element (L_{1} + … + L_{n})/2:

Λ_{W} = ℤ{L_{1}, …, L_{n}, (L_{1} + … + L_{n})/2}.

Now it can be checked, that Λ_{W} /Λ_{R} is ℤ_{2} if m = 2n + 1, ℤ_{4} if m = 2n and n is odd and ℤ_{2} ⊕ ℤ_{2} if m = 2n and n is even. The fundamental weights of 𝔰𝔬_{2n}ℂ are ω_{1} = L_{1}, ω_{2} = L_{1} + L_{2}, …, ω_{n−2} = L_{1} + … + L_{n−2} and the vectors ω_{n−1} = α = (L_{1} + … + L_{n})/2 and ω_{n} = β = (L_{1}+L_{2}+…+L_{n−1}−L_{n})/2. The fundamental weights of 𝔰𝔬_{2n+1}ℂ are ω_{1} = L_{1}, ω_{2} = L_{1} + L_{2}, …, ω_{n−1} = L_{1} + … + L_{n−1} and the vector ω_{n} = α = (L_{1} + … + L_{n})/2. The fundamental irreducible representations of the orthogonal groups can be described completely:

Proposition 4.9.

(i) representations of 𝔰𝔬_{2n}ℂ The exterior powers ∧^{k}V of the standard representation V are irreducible for k = 1, …, n − 1 and Γ_{ω_{i}} = ∧^{k}V and ∧^{n}V = Γ_{α} ⊕ Γ_{β}. The irreducible representations Γ_{α} and Γ_{β} are called spin representations.

(ii) representations of 𝔰𝔬_{2n+1}ℂ The exterior powers ∧^{k}V of the standard representation V are irreducible for k = 1, …, n and Γ_{ω_{i}} = ∧^{k}V for i = 1, …, n − 1. Γ_{α} is called the spin representation of 𝔰𝔬_{2n+1}ℂ.

5. Homogeneous Spaces, Flag Manifolds, Minimal Orbits

(see [?], 23.3)

A homogeneous space of a compact connected group G is a space X with a transitive G−action. In this section we introduce and describe the compact homogeneous spaces associated to the classical Lie groups. We begin by introducing the notion of Borel subalgebras and Borel subgroups. A choice of Cartan-subalgebra 𝔥 determines a decomposition

𝔤 = 𝔥 ⊕ (⊕_{α∈R}𝔤_{α}).

If we choose an ordering of the root system R = R^{+} ∪ R^{−}, we can associate a subalgebra

𝔟 = 𝔥 ⊕ (⊕_{α∈R^{+}}𝔤_{α})

called a Borel subalgebra. Then 𝔟 is solvable, in fact 𝔟 is a maximal solvable subalgebra.If G is a Lie group with semisimple Lie algebra 𝔤, the connected subgroup B < G with Lie algebra 𝔟 is called a Borel subgroup.

Lemma 5.1. B is a closed subgroup of G, and the quotient G/B is compact.

Proof: Consider the adjoint representation of G on 𝔤. The action of the Borel subalgebra 𝔟 obviously preserves the subspace 𝔟 ⊂ 𝔤, and, in fact, 𝔟 is just the inverse image of the subalgebra of 𝔤𝔩(𝔤) preserving this subspace: if X = ∑ X_{α} is any element of 𝔤 with X_{α} ∈ 𝔤_{α} and X_{α} ≠ 0 for some α ∈ R^{−}, we could find an element H of 𝔥 ⊂ 𝔤 with ad(X)(H) ∉ 𝔟 – any H not in the annihilator of α ∈ 𝔥^{∗} would do. B is thus (the connected component of the identity in) the inverse image in G of the subgroup of GL(𝔤) carrying 𝔟 into itself. It follows that B is closed, and the quotient G/B is contained in a Grassmannian, and hence compact. (Alternatively, we could consider the action of G on the projective space ℙ(∧^{m}𝔤) where m is the number of positive roots, and observe that B is the stabilizer of the point corresponding to the exterior product of the positive root spaces.) ⊠

In the case of the classical groups it is easy to describe the Borel subgroups and the corresponding quotients. For G = SL_{n+1}ℂ, B is the group of all upper triangular matrices in G, i.e, the automorphisms preserving the standard flag. It follows that G/B is the usual complete flag manifold, i.e, the variety of all flags

G/B = {0 ⊂ V_{1} ⊂ … ⊂ V_{n} ⊂ ℂ^{n+1}}

of subspaces with dim(V_{r}) = r. For G = SO_{2n+1}ℂ the orthogonal group of automorphisms of ℂ^{2n+1} preserving a quadratic form Q, B is the subgroup of automorphisms, which preserve a fixed flag V_{1} ⊂ … ⊂ V_{n} of isotropic subspaces with dim(V_{r}) = r. All such flags being conjugate, G/B is the variety of all such flags, i.e

G/B = {0 ⊂ V_{1} ⊂ … ⊂ V_{n} ⊂ ℂ^{n+1} : Q(V_{n}, V_{n}) = 0}.

Since B automatically preserves the flag of orthogonal subspaces, so we could also characterize G/B as the space of complete flags equal to their orthogonal complements, i.e

G/B = {0 ⊂ V_{1} ⊂ … ⊂ V_{n} ⊂ ℂ^{n+1} : Q(V_{i}, V_{2n+1−i}) = 0}.

The same holds for Sp_{2n}ℂ: the Borel subgroups B ⊂ Sp_{2n}ℂ are just the subgroups preserving a half-flag of isotropic subspaces, or equivalently a full flag of pairwise complementary subspaces; and the quotient G/B is the variety of all such flags. For G = SO_{2n}ℂ, B fixes an isotropic flag V_{1} ⊂ … ⊂ V_{n−1}, and

G/B = {0 ⊂ V_{1} ⊂ … ⊂ V_{n−1} ⊂ ℂ^{2n} : Q(V_{n−1}, V_{n−1}) = 0}.

We say, that a closed, connected subgroup P of G is parabolic, if the quotient G/P can be realized as the closed orbit of the action of G on ℙ(V) for some representation V of G, so in particular G/P is a projective algebraic variety. As we will see many famous variety is of this kind, so they are orbits at representations of a classical Lie group, and for minimal orbits we will have a strong method (The Atiyah-Bott integral formula and the Duistermaat-Heckman formula) to determine the Thom-polynomial. It follows from the proof of the previous lemma that every Borel subgroup of G is parabolic, and parabolic subgroups are those subgroups containing a Borel subgroup, so the Borel subgroups are the minimal parabolic subgroups:

Lemma 5.2. ([?], 23.47, 23.48)

(i) If B is a Borel, P is a parabolic subgroup of G, then there is an x ∈ G with B ⊂ xP x^{−1}.

(ii) If a subgroup P of G contains a Borel subgroup B, then P is parabolic.

So we can completely classify the parabolic subgroups of a simple group up to conjugacy. We may assume that P contains a Borel subgroup B, so its Lie algebra 𝔭 is a subalgebra of 𝔤 containing 𝔟 so it is of the form:

𝔭 = 𝔥 ⊕ (⊕_{α∈T} 𝔤_{α})

where T ⊂ R is generated by R^{+} and some negative simple roots. So if Σ ⊂ {positive simple roots}, T (Σ) = (ℤ{−Σ} ∩ R^{−}) ∪ R^{+} then

𝔭(Σ) = 𝔥 ⊕ (⊕_{α∈T (Σ)}𝔤_{α})

is a parabolic subalgebra, the corresponding Lie group P (Σ) is a parabolic subgroup, and we obtain in this way all the parabolic subgroups of G. We will use later the expression: a parabolic subgroup corresponding to a set of positive simple roots. (It would be better to talk about parabolic subgroup corresponding to a set of negative roots, but never mind)

Proposition 5.3. Up to conjugacy, parabolic subgroups of the simple group G are in one-to-one correspondence with subsets of the nodes of the Dynkin diagram, i.e, with subsets of the set of simple roots.

Parabolic subgroups of a classical Lie group G may be described as the subgroups preserving a partial flag in the standard representation. In particular, a maximal parabolic subgroup, corresponding to omitting one simple root (or one node in the Dynkin diagram) may be described as the subgroup of G preserving a single subspace, so G/P is the variety of these single subspaces.

Examples 5.4. For G = SL_{m}ℂ, if we omit the k^{th} simple root (the kth node in the Dynkin diagram) we get the ordinary Grassmannian G(k, m) of k-dimensional subspaces of ℂ^{m}. For Sp_{2n}ℂ, the k^{th} node of the Dynkin diagram corresponds to the Lagrangian Grassmannian of isotropic k-planes for k = 1, …, n. Similarly for G = SO_{2n+1}ℂ the kth node corresponds to the orthogonal Grassmannian of isotropic k-planes in ℂ^{2n+1}. Finally, for SO_{2n}ℂ, for k = 1, 2, …, n − 2 the kth node of the Dynkin diagram yields the orthogonal Grassmannian of isotropic k-planes in ℂ^{2n}, but there is one anomaly: either of the last two nodes gives one of the two connected components of the Grassmannian of isotropic n-planes, these are also called spinor varieties.

There is another way to realize the compact homogeneous spaces associated to a simple compact, connected group G. Let V = Γ_{λ} be an irreducible representation of G with highest weight λ, and consider the action of G on the projective space ℙV.(Every representation is a linear action, so it is an action on the corresponding projective space.) Let p ∈ ℙV be the point corresponding to the weight space of λ, i.e to the one-dimensional eigenspace, with eigenvalue λ. We have then the

Observation 5.5. The orbit G ∘ p is the unique closed orbit of the action of G on ℙV ,so this is the minimal orbit.

Proof: The point p is fixed under the Borel subgroup B, because p corresponds to a highest weight space. So the stabilizer of p ∈ ℙV is a parabolic subgroup P_{λ}. The orbit G/P_{λ} is thus compact and hence closed. Conversely, by the Borel fixed point theorem ([?]) any closed orbit of G contains a fixed point for the action of B, but p is the unique point in ℙV fixed by B. ⊠

In fact, it is not hard to say which parabolic subgroup P_{λ} is the stabilizer of this projective orbit in terms of the classification above: it is the parabolic subgroup corresponding to the subset of positive simple roots that are perpendicular to the weight λ.(??) Now, sets Σ of simple roots correspond to faces of the Weyl chamber, namely, the face that is the intersection of all hyperplanes perpendicular to all roots in Σ. We thus have a correspondence between faces of the Weyl chamber, and parabolic subgroups P, such that if V = Γ_{λ} is the irreducible representation with highest weight λ, then the unique closed orbit of the action of G on ℙV is of the form G/P, where P is the parabolic subgroup corresponding to the open face of the Weyl-chamber containing λ. In particular, weights in the interior of the Weyl chamber correspond to P_{λ} = B, and so determine the full flag manifold G/B, whereas weights on the edges give rise to the quotients of G by maximal parabolics.

Remark 5.6. We do obtain in this way all compact homogeneous spaces for G.

6. The Duistermaat-Heckman formula

Let G be a compact, connected group, T ⊂ G a maximal torus, and ρ : G ↷ V a linear action of G on the vector space V. It means that it induces an action G ↷ ℙV. Let ℙη ⊂ ℙV be a G-invariant, smooth projective variety of complex codimension c, and η = cone(ℙη) ⊂ V its cone. Then η is a G-invariant affine variety, which is homogeneous, that is, it is union of lines through the origin. Let us suppose, that G has finite number of fixed points in ℙη. The fixed points in ℙη correspond to one-dimensional eigenspaces, i.e one-dimensional representations of the action G ↷ V. This proof of the Duistermaat Heckman formula from the Atiyah-Bott integral formula is due to L. Fehér.

6.1. Connection between affine and projective Thom polynomials. Let [η]_{G} ∈ H_{G}^{2c}(V) = H_{G}^{2c}(pt) = H^{2c}(BG) be the equivariant Poincaré dual of η where working over reals ℝ, H^{∗}(BG) = ℝ[α_{1}, …, α_{r}]^{𝔚}, (r = dimT) as we have seen. We call [η]_{G} the affine Thom polynomial. Similarly [ℙη]_{G} ∈ H_{G}^{2c}(ℙV) means the equivariant Poincaré dual of ℙη, this is the projective Thom polynomial. Our next goal is to determine the connection between [η]_{G} and [ℙη]_{G}. We will use the notations introduced for equivariant characteristic classes: for example e_{G}(V) denotes the Euler class of the bundle BV → BG. By the Leray-Hirsch theorem we know, that H^{∗}(BℙV) is a module over H^{∗}(BG) in the following way: The fiber of BℙV is ℙV = ℂP^{n}, let y ∈ H^{2}(ℂP^{n}) denote the generator of the ring H^{∗}(ℂP^{n}) = R[1, y, …, y^{n−1}] then there is x ∈ H^{2}(BℙV) with j^{∗}(x^{i}) = y^{i} where j : ℙV → BℙV is the embedding of an arbitrary fiber, and H^{∗}(BℙV) is a free H^{∗}(BG) module with generators x^{i}. Now we want to determine the ring structure on H^{∗}(BℙV):

Lemma 6.1. Let w_{1}(V), …, w_{n}(V) ∈ H^{2}(BG) denote the weights of the representation G ↷ V, where n = dimV. Then

H_{G}^{∗}(ℙV) = H^{∗}(BG)[x]/(Π_{1}^{n}(x + w_{j})).

We will use the notation Γ(x) for Π_{1}^{n}(x + w_{j}).

Note: Let π : BℙV → BG be the projection map, then to be correct we should write π^{∗}w_{i} instead of w_{i} in the previous formula, but we will omit it.

Proof: Let τ denote the tautological line bundle over ℙV, then Bτ → BℙV is the “equivariant” line bundle over BℙV, so c(Bτ) = 1 + c_{1}(Bτ).

Now let π^{∗}BV denote the pull-back of the principal V -bundle

\xymatrix{π^{∗}(BV) \ar[r] \ar[d] & BV \ar[d] \\ BℙV \ar[r]^{π} & BG} .

Then

c(π^{∗}BV) = 1 + π^{∗}c_{1}(BV) + π^{∗}c_{2}(BV) + … + π^{∗}c_{n}(BV) = ∏_{i=1}^{n}(1 + π^{∗}(w_{i})).

Let Q = π ∗ BV ⊝ Bτ be the formal difference of the two bundle over BℙV, then

c(Q) = \frac{c(π^{∗}BV)}{c(Bτ)} = c(π^{∗}BV)(1 + x + x^{2} + …)

where we introduced the notation x = −c_{1}(Bτ). Now since Q is a n − 1 dimensional bundle over BℙV, we have

0 = c_{n}(Q) = ∑_{i=0}^{n} c_{n−i}(π^{∗}BV)x^{i} = ∑_{i=0}^{n} σ_{n−i}(π^{∗}w_{1}, …, π^{∗}w_{n})x^{i} = ∏_{i=1}^{n}(x + π^{∗}w_{i}).

Now let us consider the following map from the polynomial ring over H^{∗} to H^{∗}(BℙV):

q : H^{∗}(BG)[x] → H^{∗}(BℙV)
x → −c_{1}(Bτ)
α → π^{∗}α

for α ∈ H^{∗}(BG). Since restriction of c_{1}(Bτ) to any fiber gives a generator of H^{∗}(fiber), the Leray-Hirsch theorem says that q is surjective. We have just proved, that q(∏(x + w_{j})) = 0. So it remains to prove, that there are no other relations in the polynomial ring. Since we work over ℚ, it is enough to prove, that the Betti-numbers of H^{∗}(BG)[x]/ ∏(x + w_{i}) and H^{∗}(BℙV) are equal. From the spectral sequence of ℂP^{(}n − 1) → BℙV → BG we get that b^{2i}(BℙV) = ∑_{j=0}^{n−1} b^{2(i−j)}(BG). Since the coefficient of x^{n} is 1 in ∏(x + w_{i}), we have the same Betti-numbers for the other side. ⊠

Corollary 6.2. Since the main coefficient of Γ(x) is 1, every element α ∈ H^{k}(BℙV) can be written in the form α = ∑_{i=0}^{n−1} a_{i}x^{i} where a_{i} ∈ H^{k−2i}(BG).

Corollary 6.3. [ℙη]_{G} = ∑_{i=0}^{c} g_{i}x^{i} where g_{i} ∈ H^{2c−2i}(BG).

First we show how can we get the affine Thom polynomial and the degree of ℙη from the projective Thom polynomial. First here is the definition of degree:

Definition 6.4. Let X^{k} ⊂ ℙ^{n} be a closed projective variety of dimension k. Since H^{∗}(ℙ^{n}; ℤ) = ℤ[ξ]/ξ^{n+1}, there is an integer d, such that the dual class [X]_{ℙV} = d · ξ^{n−j}. This number d is called the degree of the variety X, and is denoted by deg(X).

Remark 6.5. There is a more intuitive, but less precise definition of the degree as follows: let H_{1}, …, H_{n−k} be generic hyperplanes in ℙV. Then X ∩ H_{1} ∩ H_{2} ∩ … ∩ H_{n−k} consist of finitely many points, and the cardinality of this set is the degree. The equivalence of the definitions can be shown easily: it follows from the fact that ξ = [H_{i}]_{ℙV} (i = 1, …, n − k) and [A ∩ B]_{ℙV} = [A]_{ℙV} ∪ [B]_{ℙV} for “generic” A and B.

Lemma 6.6. [?] If [ℙη]_{G} = ∑_{i=0}^{c} g_{i}x^{i} = p(x) then g_{0} = [η]_{G} ∈ H^{2c}(BG), and g_{c} = deg(ℙη) · 1, where 1 ∈ H^{0}(BG) is the generator.

Proof: We don’t prove the degree part of the assertion. Let us consider the following diagram:

\xymatrix{ & V \\ ℙV & V \ 0 \ar[l]^{π} \ar[u]_{i}}

where i is the embedding, π is the natural projection. Now applying the functor H_{G}^{∗} to this diagram we get

\xymatrix{ & H^{∗}(BG) \ar[dl] \ar[d]^{i_{G}^{∗}} \\ H_{G}^{∗}(ℙV) = H^{∗}(BG)[x]/Γ(x) \ar[r]^{π_{G}^{∗}} & H_{G}^{∗}(V \ 0)}

where π_{G}^{∗}([ℙη]_{G}) = [η \ 0]_{G} = i_{G}^{∗}([η]).

Now π_{G}^{∗}(x) = π_{G}^{∗}(c_{1}(τ)) = 0 since x = −c_{1}(Bτ), where Bτ → BℙV is the tautological line bundle over BℙV, and π^{∗}(τ) is trivial line bundle over V \ 0 hence π^{∗}(c_{1}(τ) = 0, so the equivariant Chern class goes to 0 as well, so π_{G}^{∗}(p(x)) = π_{G}^{∗}(p(0)). But i_{G}^{∗} is isomorphism for ∗ < n from the Gysin sequence, so π_{G}^{∗}(p(0)) = p(0) under this isomorphism using the commutativity of the diagramm, and the lemma is proved.

After this lemma we can think that projective Thom polynomials contain more information than the affine ones, but this is not the truth. In fact a simple substitution into the affine Thom polynomial provides the projective Thom polynomial in the following way, see [?] Let m : U(1)^{n} → G be a coordinated maximal torus of G and let α_{i} are the corresponding weights, i.e the generators of H^{∗}(BG). By the Borel theorem (see ??) [η]_{G} = T p(η) is a polynomial in variables α_{i} and [ℙη]_{G} = T p(ℙη) is a polynomial in variables α_{i} and ξ. We assume that the image of ρ contains the scalars, i.e. there is a homomorphism φ : GL(1) → G and a non-zero integer q such that ρ∘φ(λ)v = λ^{q}v for v ∈ V, λ ∈ GL(1). We assume that Imφ|_{U(1)} ⊂ Im m, so φ factors through U(1)^{r}, that is, we have a homomorphism φ̃ : U(1) → U(1)^{n} such that φ|_{U(1)} = m ∘ φ̃, i.e. the following diagram is commutative:

\xymatrix{GL(1) \ar[r]^{φ} & G \\ U(1) \ar[u]_{i} \ar[r]^{φ̃} & U(1)^{r} \ar[u]^{m}}

The homomorphism φ̃ must be of the form φ̃(z) = (z^{ω_{1}}, …, z^{ω_{n}}) where z ∈ U(1) and ω_{i} are integers. The choice of φ is not unique.

Theorem 6.7. ([?]) Let G → GL(V) be a representation of the Lie group G, such that the image of ρ contains the scalars. Let α_{1}, q, ω_{i}, η as above. Then

T p(ℙη)(α_{1}, …, α_{n}, ξ) = T p(η)(α_{1} + \frac{ω_{1}}{q} ξ, …, α_{n} + \frac{ω_{n}}{q} ξ).

Proof The trick is that we relate the projective Thom polynomial to an affine Thom polynomial for a different group. Let us consider the bundle χ : Hom(ℂ, V) → Hom(τ, π^{∗}BV) → B_{G}ℙV. (here we use the notations of ??) This bundle has G̃ = GL(1)×G as its structure group, with the action on Hom(ℂ, V) by ρ̃ = Hom(ρ^{1}, ρ) where ρ^{1} is the standard representation of GL(1) on ℂ. Since Hom(ℂ, V) is isomorphic to V in a natural way, we can think as G̃ acting on V. Moreover the G̃-invariant subsets of Hom(ℂ, V) are the same as the G-invariant subsets of V, if we use this isomorphism. The bundle χ has a canonical section σ, since a fiber of τ is a part of the corresponding fiber π^{∗}V over the same point x of B_{G}ℙV, and σ(x) is this embedding. By definition, σ^{−1}(η) = B_{G}ℙη. So if f : B_{G}ℙV → BG̃ denotes the classifying map of χ then f^{∗}T pρ̃(η) = T p_{ρ}(ℙη). Now it is easy to see, that H^{∗}(BG̃) ≅ H^{∗}(BG)[ξ_{1}] where ξ_{1} = −c_{1}(π_{1}^{∗}τ_{1}), π_{1} : BG̃ → BGL(1) is the projection and τ_{1} is the tautological line bundle over BGL(1). Moreover f^{∗} : H_{G}^{∗}(ξ_{1}) → H_{G}^{∗}(ℙV) is a H_{G}^{∗}-module homomorphism which maps ξ_{1} to ξ, so it s simply the factorization with the relation Γ(ξ) = ∏(ξ + ω_{i}). So we must concentrate on calculating T pρ̃(η). To do that we choose the maximal torus of G̃ of the form m̃(t_{0}, t_{1}, …, t_{n}) = (t_{0}, m(t_{1}, …, t_{n})). The key observation is that restricted to these maximal tori the representation ρ̃ almost factors through ρ. We define a homomorphism κ : U(1)^{n+1} → U(1)^{n} in the following way:

κ(t_{0}, t_{1}, …, t_{n}) = (t_{0}^{ω_{1}}t_{1}^{q}, …, t_{0}^{ω_{n}}t_{n}^{q}) = φ̃(t_{0})(t_{1}^{q}, …, t_{n}^{q}).

Then

ρ ∘ κ(t_{0}, …, t_{n}) = ρ(φ̃(t_{0})(t_{1}^{q}, …, t_{n}^{q})) = (t_{0} · ρ(t_{1}, …, t_{n}))^{q} = (ρ̃ ∘ m̃(t_{0}, …, t_{n}))^{q}.

We will use the following proposition twice:

Proposition 6.8. Let h : K → G be a homomorphism of Lie groups and σ = ρ ∘ h. Then

1. if η ⊂ V is ρ-invariant, then η is σ-invariant too.

2. T p_{σ}(η) = (Bh)^{∗}T p_{ρ}(η) where Bh : BK → BG is induced by h.

This proposition is an obvious consequence of the fact, that the universal σ-bundle is the pull back of the universal ρ-bundle via Bh. Applying this twice: first to h = κ, (Bh)^{∗}(α_{i}) = qα_{i} + ω_{i}ξ, then to h(x) = x^{q}, (Bh)^{∗}(α_{i}) = qα_{i} finishes the proof of ??.

Corollary 6.9. The degree of the projective variety ℙη comes from a simple substitution:

deg(ℙη) = T p(η) (\frac{ω_{1}}{q} , …, \frac{ω_{n}}{q} ) .

We will see one application of this theorem for Grassmannians in ??

6.2. Push-forward and Thom-polynomials. A good reference for this and the next subsection is [?]. Let N^{n}, M^{m} be compact, oriented manifolds of dimension n, m and f : N → M be a map. Then we can define the “push-forward”, or Gysinmap of H^{∗}

f_{!} : H^{∗}(N) → H^{∗+m−n}(M)

with the aid of Poincaré duals:

\xymatrix{H^{∗}(N) \ar[d]^{P D} \ar[r]^{f_{!}} & H^{∗+m−n}(M) \\ H_{n−∗}(N) \ar[r]^{f_{∗}} & H_{n−∗}(M) \ar[u]^{P D}}

so f_{!}(α) = P D(f_{∗}P D(α)). More general we can define “push-forward” for proper maps of spaces. (proper means, that the pre-image of compact set is compact) This “push-forward” has the following properties:

(i) It is functorial: (f ∘ g)_{!} = f_{!} ∘ g_{!}

(ii) It is a homomorphism of H^{∗}(M)-modules,i.e

f_{!}(αf^{∗}β) = (f_{!}α)β for α ∈ H^{∗}(N), β ∈ H^{∗}(M)

this is called adjunction formula.

(iii) If f is a fibering, then f_{!} corresponds to integration over the fiber.

(iv) If f : N ↪ M is embedding of a submanifold, then f_{!}(1) = P D(N ↪ M), the Thom-class of the normal bundle of N in M, so f^{∗}(f_{!}(1)) = e(ν_{f}), the Euler-class, and in general f_{!} = j^{∗} ∘ φ where φ is the Thom-isomorphism:

\xymatrix{… \ar[r] & H^{∗−1}(M − N) \ar[r]^{δ} & H^{∗}(M, M − N) \ar[r]^{j^{∗}} \ar[d]^{φ} & H^{∗}(M) \ar[r] & … \\  &  & H^{∗−(m−n)}(N)}

This “push-forward” can be defined almost word-for-word in the equivariant situation. Let G be a compact group, N, M be G-spaces, and g : N → M G-equivariant map. In this situation we can define the “equivariant push-forward” map

g_{!}^{G} : H_{G}^{∗}(N) → H_{G}^{∗+m−n}(M).

The map g induces a map Bg : B_{G}(N) → B_{G}(M), but these spaces are already not manifolds, and not compact, so the definition with Poincaré dual does not work. Fortunately by the graph construction every map g can be factored into an inclusion followed by a product projection in the following way:

\xymatrix{N \ar[r]^{i} & N × M \ar[r]^{π} & M \\ n \ar[r] & (n, g(n)) \ar[r] & g(n)}

So it is enough define the “push-forward” for inclusions and fibering.

(i) If g : N ↪ M is inclusion, so N is G-equivariant submanifold of M, then since G is compact, there is tubular neighborhood U of N in M, which is Gequivariant, so B_{G}U is tubular neighborhood of B_{G}(N) in B_{G}(M), so we can define g_{!} = j^{∗} ∘ φ as above, with the Thom-isomorphism.

(ii) If g : N → M is a fibering, the induced map Bg : B_{G}N → B_{G}M is one also with compact oriented fiber, and integration over the fiber is well-defined in any fibering with an oriented compact manifold as fiber, so we can define g_{!}^{G} in this way.

The “equivariant push-forward” satisfies moreover:

(iii) g_{!}^{∗} preserves the H^{∗}(BG)-module structure.

(iv) The special case of (ii) is when M = pt: if π : N → pt then π_{!}^{G} corresponds to

integration over the fiber in the fibering B_{G}M → BG.

Notation: We introduce a special notation for the “equivariant push-forward map” in the previous case (iv), when M = pt. Let

∫_{N} α = π_{!}^{G}(α) ∈ H^{∗}(BG), α ∈ H_{G}^{∗}(N)

according to the fact, that π_{!}^{G} is integration over the fiber. So integration is a map:

∫_{N} : H_{G}^{∗}(N) → H^{∗−n}(BG)

(v) As we mentioned in ?? there is a natural map i^{∗} : H_{G}^{∗}(X) → H^{∗}(X) from equivariant theory to ordinary theory which corresponds to the inclusion i : X → B_{G}X of X as the fiber over a point of BG. Let N, M be n and mdimensional manifolds, respectively, and f : N → M be a proper map. Then we have the following commutative diagram:

\xymatrix{H_{G}^{∗}(N) \ar[r]^{i_{N}^{∗}} \ar[d]^{f_{!}^{G}} & H^{∗}(N) \ar[d]^{f_{!}} \\ H^{∗+m−n}(BG) \ar[r]^{i_{M}^{∗}} & H^{∗+m−n}(M)}

Lemma 6.10. Let ∑_{i=0}^{n−1} a_{i}x^{i}, (a_{i} ∈ H^{k−2i}(BG)) be an arbitrary element of H^{k}(BℙV) = H_{G}^{∗}[x]/Γ(x), see ??. Then

∫_{ℙV} ∑_{i=0}^{n−1} a_{i}x^{i} = a_{n−1}.

Proof: Since the integration ∫_{ℙV} : H_{G}^{∗}(ℙV) → H^{∗−2n+2}(BG) lower the dimension by 2n − 2, it sends 1, x, …, x^{n−2} to 0. The remaining x^{n−1} goes to 1 ∈ H^{0}(BG), which can be seen from the following commutative diagram, which is a special case of (v):

\xymatrix{H_{G}^{2n−2}(ℙV) \ar[r]^{i_{ℙV}^{∗}} \ar[d]^{π_{!}^{G}} & H^{2n−2}(N) \ar[d]^{π_{!}} \\ H^{0}(BG) \ar[r]^{i_{pt}^{∗}} & H^{0}(pt)} .

Now i_{ℙV}^{∗} (x^{n−1}) is the generator of H^{2n−2}(ℙV), and by the definition of π_{!} in the non-equivariant case (i_{ℙV}^{∗} ∘ π_{!})(x^{n−1}) = 1 ∈ H^{0}(pt). The diagram is commutative, and i_{pt}^{∗} : H^{0}(BG) → H^{0}(pt) is isomorphism, so π_{!}^{G}(x^{n−1}) = 1 ∈ H^{0}(BG). ⊠

We have seen at the proof of Lemma ?? that x = −c_{1}(Bτ) where τ is the tautological line bundle over ℙV, and Γ(x) = ∏(x + π^{∗}w_{i}) where w_{i} are the weights of the representation V, so

Γ(x) = x^{n} + c_{1}(π^{∗}BV) + … + c_{n−1}(π^{∗}BV) + e(π^{∗}BV),

so we can define

q = −\frac{π^{∗}BV}{x} = x^{n−1} + c_{1}(π^{∗}BV)x^{n−2} + … + c_{n−1}(π^{∗}BV) ∈ H^{2n−2}(BℙV)

The following lemma gives the connection between the projective, and affine Thompolynomial:

Lemma 6.11.

[η]_{G} = ∫_{ℙV} q · [ℙη]_{G}

Proof: Let [ℙη]_{G} = ∑_{i=1}^{c} g_{i}x^{i} where g_{i} ∈ H^{2c−2i}(BG) be as above. Then

q · [ℙη]_{G} = q(∑_{i=1}^{c} g_{i}x^{i}) = g_{0}q + (∑_{i=1}^{n−1} g_{i}x^{i−1})e(π^{∗}BV),

so using the previous theorem we get

∫_{ℙV} q · [ℙη]_{G} = ∫_{ℙV} g_{0}q + ∫_{ℙV} (∑_{i=1}^{n−1} g_{i}x^{i−1})e(π^{∗}BV) = g_{0} = [η]_{G}.⊠

Lemma 6.12. Let Bi : Bℙη → BℙV denote the map induced by the inclusion ℙη ↪ ℙV. Then

∫_{ℙV} q · [ℙη]_{G} = ∫_{ℙη}(Bi)^{∗}q.

Proof: We have the following diagram:

\xymatrix{ℙη \ar[r]^{i} \ar[dr]_{π^{ℙη}} & ℙV \ar[d]^{π^{ℙV}} & B_{G}ℙη \ar[r]^{Bi} \ar[dr]_{Bπ^{ℙη}} & BℙV \ar[d]^{Bπ^{ℙV}} \\  & pt &  & BG}

Now using that the push-forward is module-homomorphism:

∫_{ℙη}(Bi)^{∗}q = ∫_{ℙV} i_{!}^{G}((Bi)^{∗}q · 1) = ∫_{ℙV} q · i_{!}1 = ∫_{ℙV} q · [ℙη]_{G}.⊠

6.3. The Atiyah-Bott localization theorem, and integral formula. The standard reference is [?]. From now on, we will work with complex coefficients, so H_{G}^{∗} means equivariant cohomology with complex coefficients. When G = T^{l} is a torus of dimension l, then H_{G}^{∗} is the polynomial ring ℂ[u_{1}, …, u_{l}] and this is not a principal ideal domain. As a consequence, modules over this ring no longer have so simple structure as a sum of a free and a torsion module. However we can define the torsion sub-module and the rank. This can be done by passing to the field of fractions ℂ(u_{1}, …, u_{k}).

We can define the support of a finitely generated module in the following way: Let H be a module over ℂ[u_{1}, …, u_{l}] its support is the subset of ℂ^{l} defined by:

SuppH = ⋂_{f} V_{f} over all f ∈ ℂ[u_{1}, …, u_{l}] with fH = 0

where V_{f} is the hypersurface f(u) = 0. Thus a free module has the whole space ℂ^{l} as a support, while the support of a torsion module indicates which “primes” occur in the torsion. A related notion is that of localization. If f ∈ ℂ[u_{1}, …, u_{l}] is any non-zero polynomial, localizing to the open set U_{f} = ℂ^{l} − V_{F} means that we allow ourselves to divide by powers of f. More formally we form the ring ℂ[u_{1}, …, u_{l}]_{f} consisting of all rational functions with denominators power of f, and consider the corresponding module

H_{f} = H ⊗_{ℂ[u_{1},... ,u_{l}]} ℂ[u_{1}, …, u_{l}]_{f}

for any module H over ℂ[u_{1}, …, u_{l}]. This process kills torsion supported in V_{f}, i.e.

SuppH ⊂ V_{f} ⇒ H_{f} = 0.

Let T act smoothly on the compact manifold M with F as the set of fixed points. Since the action of T on F is trivial, we have B_{T} F ≃ BT × F so H_{T}^{∗} (F) = H_{T}^{∗} ⊗ H^{∗}(F), so H_{T}^{∗} (F) is a module over H_{T}^{∗}. As we have seen for the inclusion i : F ↪ M the push-forward map i_{!} : H_{T}^{∗} (F) → H_{T}^{∗} (M) is a H_{T}^{∗} (M)-module homomorphism, and

i^{∗}i_{!}1 = e_{T} (ν_{F})

so i^{∗}i_{!} is multiplication in H_{T}^{∗} (F) by the equivariant Euler-class of the normal bundle ν_{F}:

i^{∗}i_{!}(i^{∗}α) = i^{∗}i_{!}(i^{∗}α · 1) = i^{∗}(α · i_{!}1) = (i^{∗}α)e_{T} (ν_{F}).

We will show, that e_{T} (ν_{F}) is be invertible in the localized module

H_{T}^{∗} (F)_{f} = (H_{T}^{∗})_{f} ⊗ H^{∗}(F)

where f ∈ ℂ[u_{1}, …, u_{l}] = H_{T}^{∗} is a suitable polynomial. We can give this polynomial explicitly in the following way. Let F = ∪P be the decomposition of F into its connected components, and we may consider separately each e_{T} (ν_{P}). Since terms of positive degree in H^{∗}(P) are nilpotent, an element in H^{∗}(P) is invertible provided its component in H^{0}(P) is nonzero. Hence e_{T} (ν_{P}) will be invertible in some localization H_{T}^{∗} (P)_{f} provided its component in H_{T}^{∗} ⊗ H^{0}(P) is a non-zero polynomial, and we choose this polynomial as f. But this component is determined by restricting to any point p of P. The action of T on the fiber ν_{p} has no fixed vectors, because the only fixed directions are tangential to P, so ν_{p} decomposes as a direct sum of irreducible one-dimensional representations of T, and if a_{j} ∈ H^{∗}(BT) denotes the weights of the representation ν_{p} then e_{T} (ν_{p}) = ∏_{j} a_{j} ≠ 0, because if a_{i} was 0, it would mean, that this weight space is a fix one-dimensional subspace, a contradiction. If we denote this non-zero polynomial by f_{P} it follows that i^{∗}i_{!} becomes invertible after localizing with respect to

f_{F} = ∏_{P} f_{P} .

Working over such a localized ring (or over the full field of rational functions) we therefore see that

Q = ∑_{P} \frac{i_{P}^{∗}}{e_{T} (ν_{P})}

is inverse to i_{!}^{F} : H_{T}^{∗} (F) → H_{T}^{∗} (M). (This is the same as i_{!}, but we need the upper index to distinguish this map from i_{!}^{P} : H_{T}^{∗} (P) → H_{T}^{∗} (M).) Thus for any φ ∈ H_{T}^{∗} (M) we have after localizing

φ = i_{!}^{F} Qφ = ∑_{P} \frac{i_{!}^{P} i_{P}^{∗} φ}{e_{T} (ν_{P})}.

Applying the push-forward to point π_{!}^{M} : H_{T}^{∗} (M) → H_{T}^{∗} to both sides, we get

π_{!}^{M}φ = ∑_{P} π_{!}^{M}i_{!}^{P} \frac{i_{P}^{∗} φ}{e_{T} (ν_{P})}

Now using the functoriality of push-forward, that is π_{!}^{M} ∘ i_{!}^{P} = (π^{M} ∘ i^{P})_{!} = π_{!}^{P} we get the integration formula:

Theorem 6.13. Integration formula[?] Let M be a compact oriented manifold with a T -action, F is the set of fixed points of the action and P are the connected components of F. Let i_{P} : P ↪ M be the inclusion of P, i_{P}^{∗} : H_{T} (M) → H_{T} (P), φ ∈ H_{T}^{∗} (M). Then e_{T} (ν_{P}) ≠ 0 and

∫_{M} φ = ∑_{P} ∫_{P} \frac{i_{P}^{∗} φ}{e_{T} (ν_{P})}.

6.4. The Duistermaat-Heckman formula for Thom-polynomials. Here we present the proof of the Duistermaat-Heckman formula from the integration formula, this is due to L. Fehér. We will use the integration formula in the following situation. Let T be a torus, and T ↷ V a linear action of T on the vector space V. (In most cases T is the maximal torus of a compact, connected Lie-group G.) It means that it induces an action G ↷ ℙV. Let us suppose that the weights of V are different, then ℙV has finitely many fixed points, which correspond to weight spaces of V, so the set F of fixed points in this situation is a finite union of single points. Let η be an orbit of the action, such that ℙη is a smooth variety. Then using the notations of the following commutative diagram of inclusions

\xymatrix{f ∈ F ∩ ℙη \ar[r]^{i_{f}} \ar[d]_{i_{f⊂ℙη}} & ℙV \\ ℙη \ar[ur]_{i_{ℙη}}}

we get by the integral formula

[η]_{T} = ∫_{ℙη} i_{ℙη}^{∗}q = ∑_{f∈ℙη∩F} \frac{i_{f⊂ℙη}^{∗}i_{ℙη}^{∗}q}{e_{T} (ν_{f})} = ∑_{f∈ℙη∩F} \frac{i_{f}^{∗}q}{e_{T} (ν_{f})}.

where i^{∗} always means the corresponding map between equivariant cohomologies. Since f is a point, ν_{f} = T_{f}ℙη, so e_{T} (ν_{f}) = e_{T} (T_{f}ℙη). Recall, that

q = \frac{e(π^{∗}BV)}{x} = ∑_{i=0}^{n−1} c_{n−i}(π^{∗}BV)x^{i−1},

and x = c_{1}(τ), so i_{f}^{∗}x = c_{1}(i_{f}^{∗}τ) = w_{f}, where w_{f} is the weight corresponding to f, which is a weight-space, and i_{f}^{∗}e(π^{∗}BV) = e_{T} (V). So

i_{f}^{∗}q = \frac{e_{T} (V)}{w_{f}}

and we get the desired formula

Theorem 6.14. Duistermaat-Heckman formula Let T^{k} ↷ V be a linear action of T on the vector space V, ℙη ⊂ ℙV a T -invariant smooth variety, η = coneℙη,and B_{T} η ⊂ B_{T} V represents a cohomology element of H^{∗}(BT) = ℚ[α_{1}, …, α_{n}]. Let us suppose that the action T ↷ ℙV has a finite number of fixed points, F denote the set of these, and the weight corresponding to f ∈ F is w_{f}. Then

T p(η) = [η]_{T} = e_{T} (V) ∑_{f∈ℙη∩F}(e_{T} (T_{f}ℙη)w_{f})^{−1}

Remark 6.15. We don’t know any appearance of this theorem in the literature. The formula was communicated to us by A. Knutson. The right hand side of the Duistermaat Heckman formula is an element of the field of fractions of the cohomology ring H_{T}^{∗}, but we know a priori, that this is simply a symmetric polynomial, so it is a polynomial of the basic symmetric polynomials which are the Chern classes. If T ⊂ G is a maximal torus of a compact, connected group, then [η]_{T} = [η]_{G}.

7. Calculation of Thom polynomials for homogeneous spaces

7.1. The method. We can use the Duistermaat Heckman formula to compute Thom polynomials of homogeneous spaces, which appear when G is a Lie group, T ⊂ G is a maximal torus, and V is an irreducible representation of G. As we have seen in Chapter ?? we can get all compact homogeneous spaces of G as the orbit of a highest weight vector at the action of G on ℙV, where V is an irreducible representation of G, so every homogeneous G-space is of the form G/P_{λ} where P_{λ} is the parabolic subgroup corresponding to the highest weight λ. So we will use The Duistermaat Heckman formula when T ⊂ G is maximal torus, ℙη is an orbit of G, (then it is invariant subset of T) and this will be a smooth orbit, so it will define a cohomology element. The fixed points under the projective action of T are precisely the weight spaces, which will be different, so all the assumptions of the theorem stand. So let V be an irreducible representation of G, λ is a highest weight corresponding to an ordering of the weight lattice, ℙη its orbit in ℙV. Then e_{T} (V) is simply the product of the weights of V, it remains two problems: (i) to determine which of the weights are in ℙη; (ii) to compute e_{T} (T_{f}ℙη).

Theorem 7.1. The weights of V which are in the orbit ℙη are the weights of the orbit of λ under the action of the Weyl group on the weight lattice, so

F ∩ ℙη = 𝔚 ∘ λ.

These are exactly the highest weights of the representation.

Proof: From the definition and properties of the Weyl group.

The orbit of one highest weight under the action of the Weyl group consist of precisely all the highest weights of the representation. To solve the second problem, we observe that the action of a Lie group on T_{f}(ℙη), where f ∈ F is a fix point of the action, is the infinitesimal action of the Lie group at f, so T_{f}(ℙη) is a representation of the corresponding Lie algebra, namely, the adjoint representation. Precisely, let B_{f} be the stabilizer of f, which is a Borel subgroup of G, and X = G/B_{f} is the compact homogeneous space. If V_{λ} ∈ ℙV is the highest weight space, then B_{V_{λ}} = P_{λ}, and for other fixed points, which are images of v_{λ} under the Weyl group, B_{f} will be conjugate to P_{λ}. Now the action of G on G/B_{f} is given by b(gB_{f}) = bgB_{f} = bgb^{−1}B_{f}, so it coincides with conjugation, therefore the action of G on T_{e}(G/B_{f}) is the adjoint action. Now T_{e}(G/B_{f}) = 𝔤/𝔟_{f}, so we have proved the

Theorem 7.2. T_{f}(G/B_{f}) = 𝔤/𝔟_{f}, where the equality means the isomorphism of the two representation, where on the right hand side there is the adjoint representation.

The following well-known lemma helps us to determine the Lie algebra 𝔟_{f} of the stabilizer group of the fix point f:

Lemma 7.3. Let B_{f} denote the stabilizer group of the fix point f, which is a Borel subgroup. Then

Lie(B_{f}) = {H ∈ 𝔤 : H(f) = 0} = Ann(f)

where H(f) is the action of H on f. (V is a representation of 𝔤 too.)

So our exercise is to compute the annihilators of the highest weight spaces of the representation, which is now not a difficult problem, since for a highest weight space V_{λ} we we have seen in Chapter ?? how can we describe the parabolic subgroup, which is the stabilizer:

Proposition 7.4. LieP_{λ} corresponds to Σ = {𝔤_{α}}_{B(α,λ)=0} ⊂ R^{+}, so

T_{V_{λ}}ℙη = 𝔤/LieP_{λ} = ℂ{𝔤_{α} ∈ R^{−} : B(α, λ) < 0}.

Since we know how to compute Killing form in most cases, we can write up the DH formula easily. Let we see some examples.

7.2. Thom polynomials of Grassmannians.

7.2.1. The ordinary Grassmannians.

We can embed the ordinary Grassmannian manifold Gr(n, k), the k-dimensional complex subspaces of ℂ^{n} into ℙ ∧^{k} ℂ^{n} by the usual Plücker embedding:

Gr(n, k) ↪ ℙ ∧^{k} ℂ^{n}
ℂ < v_{1}, …, v_{k} >→ [v_{1} ∧ … ∧ v_{k}]

Now as we have seen ∧^{k}ℂ^{n} is an irreducible representation of SL_{n}ℂ, and so it is an irreducible representation of the corresponding Lie algebra 𝔰𝔩_{n}ℂ, with highest weight ω_{k} = L_{1} + … + L_{k}, the kth fundamental weight of the weight lattice, so this is the representation Γ_{0,... ,0,1,0,... ,0} with 1 at the ith column. Gr(n, k) is the orbit of the highest weight space in ℙ ∧^{k} ℂ^{n}. Let l_{ω_{i}} denote the highest weight space. The negative roots are R^{−} = {L_{i} − L_{j}}_{i>j}, and by proposition ?? we should determine the negative roots with

B(L_{1} + … + L_{k}, L_{i} − L_{j}) < 0.

Now as we have seen

B(L_{i}, L_{j}) = \{ \begin{matrix}\frac{−1}{(2n^{2})} & i ≠ j \\ \frac{n−1}{(2n^{2})} & i = j \end{matrix}

so we get that for i > j

B(L_{1} + … + L_{k}, L_{i} − L_{j}) = \{ \begin{matrix}\frac{−1}{(2n)} & if & j ≤ k < i \\ \frac{n−1}{(2n^{2})} & if & j < i ≤ k or k < j < i \end{matrix} .

Hence as we have seen, the tangent space of Gr(n, k) at the fix point l_{ω_{k}} is a representation of 𝔰𝔩_{n}ℂ with weights {L_{i} − L_{j}}_{j≤k<i}, so we get using the notation L_{i} for the Chern root α_{i} ∈ H^{2}(BT):

e_{T} (T_{l_{ω_{k}}} Gr(n, k)) = ∏_{j≤k<i}(L_{i} − L_{j}).

As a consequence we derive the dimension of Gr(n, k), it is k(n − k), hence the codimension of Gr(n, k) in ∧^{k}ℂ^{n} (which is the degree of the Thom-polynomial) is \binom{n}{k} − k(n − k) − 1. We know, that the the fix points are the images of the weight space l_{ω_{k}} under the Weyl group. The Weyl group simply permutes the roots, so in a general fix point we must write L_{i_{1}} + … + L_{i_{k}} instead of L_{1} + … + L_{k}, and we get the Thom polynomial of the cone of the Grassmannian, which is denoted by cGr(n, k)

Example 7.5.

T p(cGr(n, k)) = ∏_{\begin{matrix}I∈\binom{n}{k} \\ I=\{i_{1}<...<i_{k}\} \end{matrix}}(L_{i_{1}} + L_{i_{2}} + … + L_{i_{k}})·

· ∑_{\begin{matrix}I∈\binom{n}{k} \\ I=\{i_{1}<...i_{k}\} \end{matrix}} \frac{1}{(L_{i_{1}} + L_{i_{2}} + … + L_{i_{k}}) ∏\begin{matrix}i∉I \\ j∈I \end{matrix}(L_{i} − L_{j})}

Here I ⊂ \binom{n}{k} means that I runs over the k-element subsets of the index set.

Unfortunately we have not succeed to determine the polynomial form of the right side of the DH formula for n > 7 even with the aid of computers. It is possible to give a not too simple formula for the degree of the Grassmannians using theorem ??. The substitution in the formula does not work directly, since substituting α_{i} = 1/k the denominator will be zero. But with a little trick we reach our goal: substitute α_{i} = \frac{(1+iε)}{k} and then take the limit ε → 0. After writing the formula in common denominator form we see, that both the denominator and nominator tends to 0 in the order of \binom{n}{2}, so applying the l’Hospital rule \binom{n}{2} times, and then substituting ε = 0 we get the following with the notation I for the set of k-element subsets of [n], and d = k(n − k):

deg(Gr(n, k) = \begin{matrix}∑ \\ ( & {J ∈I \atop d} & ) \end{matrix} ( ∏_{J∈J} (∑_{j∈J} j) ∑_{I∉J} (∏_{\begin{matrix}i∈I \\ j∉I \end{matrix}}(j − i)^{−1}))

There is a well-known formula for this degree, see [?]:

deg(Gr(n, k) = (k(n − k))! ∏_{i=0}^{k−1} \frac{i!}{(n − k + i)!}.

It is now a good exercise to prove that this is the same as the previous formula, unfortunately we have not succeeded to do it. These are the bad news. The good is that from the Duistermaat Heckman formula it is possible to compute degrees of other symmetric varieties. For example L. Fehér has generalized this technique to compute degrees of duals of Grassmannians (which is in fact the same as the Thom-polynomial itself).

The next example shows a bit more simple case, when we can explicitly determine the Thom polynomial.

7.2.2. The Veronese variety.

For any (complex) vector space V and any positive integer n we have a natural map called the Veronese embedding

ℙV ↪ ℙ(Sym^{k}V)
[v] → [v^{k}]

so if V is n-dimensional this is a smooth embedding ℂP^{n−1} ↪ ℂP^{\binom{n+k−1}{k−1}}. If V is the standard representation of SL_{n}ℂ then we have seen that Sym^{k}V = Γ_{k,0,... ,0} is an irreducible representation with highest weight λ = kL_{1}, and the orbit of a highest weight vector in ℙ(Sym^{k}V) is the image of the Veronese embedding, called the Veronese variety, this will be denoted by V(n, k). Now

B(kL_{1}, L_{i} − L_{j}) = \{ \begin{matrix}0 & 1 < j < i \\ −\frac{k}{2n} & 1 = j < i \end{matrix}

and from Proposition ?? the Lie algebra of the stabilizer subgroup of λ

LieP_{λ} = {R^{+}, L_{k} − L_{1} (k = 2, …, n}, e(T_{kL_{1}}(V(n, k)) = ∏_{i=2}^{n}(L_{i} − L_{1}).

Now by Proposition ?? and Theorem ?? and using that the Weyl group is simply σ_{n} we get the Thom polynomial of the Veronese embedding:

T p(V(n, k)) = ∏_{1≤i_{1}≤i_{2}...≤i_{k}}(L_{i_{1}} + … + L_{i_{k}})·

· ∑_{i=1}^{n} \frac{1}{kL_{i}(L_{1} − L_{i}) … (L_{i−1} − L_{i})(L_{i+1} − L_{i}) … (L_{n} − L_{i})}

Now a little computation shows that

∑_{i=1}^{n} \frac{1}{L_{i}(L_{1} − L_{i}) … (L_{i−1} − L_{i})(L_{i+1} − L_{i}) … (L_{n} − L_{i})} = \frac{1}{L_{1} … L_{n}} .

So the Thom polynomial of the cone is the following:

Example 7.6.

T p(cV(n, k)) = ∏_{\begin{matrix}1≤i_{1}≤i_{2}...≤i_{k} \\ i_{1}≠i_{k} \end{matrix}}(L_{i_{1}} + … + L_{i_{k}}).

7.2.3. The orthogonal Grassmannian, and the spinor variety.

Next consider the irreducible representation ∧^{k}V = Γ_{ω_{k}} of SO_{2n}ℂ where V is the standard representation.(1 ≤ k ≤ n−2) The highest weight of this is the fundamental weight λ = ω_{k} = L_{1} + … + L_{k}, and the corresponding homogeneous space, which is the orbit of this highest weight space in ℙ ∧^{k} V is the smooth manifold of kdimensional orthogonal Grassmannian, which is the set of k-dimensional isotropic subspaces of V for the given bilinear form on V (see ??) we denote it by Gr_{o}(n, k). These orthogonal Grassmannians are therefore minimal orbits, the stabilizer of the highest weight space is a parabolic subgroup of SO_{2n}ℂ. The negative roots are R^{−} = {−L_{i} − L_{j}}_{i>j} ∪ {L_{i} − L_{j}}_{i>j} and

B(L_{1} + L_{2} + … + L_{k}, −L_{i} − L_{j})_{i>j} = \{ \begin{matrix}0 & k < j < i \\ −1/(n − 1) & j ≤ k < i \\ −2/(n − 1) & j < i ≤ k \end{matrix}

so B(λ, −L_{i} − L_{j}) < 0 iff {1, 2, …, k} ∩ {i, j} ≠ 0, and we get in the same way that B(λ, L_{i} − L_{j}) < 0 iff j ≤ k < i, so {1, 2, …, k} ∩ {i, j} = j. Now again by Proposition ??

e(T_{λ}Gr_{o}(n, k)) = ∏_{j<i≤k}(L_{i} + L_{j}) ∏_{j≤k<i}(L_{j} + L_{i})(L_{j} − L_{i}).

The Weyl group in this case permutes the L_{i}s, and changes the sign of even of them. So a general element of W takes L_{s} to ε_{i_{s}}L_{i_{s}} where ε_{i} = ±1, |{i : ε_{i} = −1}| = even. The image of ω_{k} = L_{1} + … + L_{k} under this action is ε_{i_{1}}L_{i_{1}} + ε_{i_{2}}L_{i_{2}} + … + ε_{i_{k}}L_{i_{k}}.

Remark 7.7. Note that L_{1} + … + L_{k} can have the same image under different elements of the Weyl group, since we can permute the weights L_{i_{k+1}}, …, L_{i_{n}} arbitrarily after saying the images of L_{1}, …, L_{k}. But e(T_{L_{i_{1}}+...+L_{i_{k}}} Gr_{o}(n, k)) will be the same at every two such element of the Weyl group, despite of the fact, that in the expression e(T_{L_{i_{1}}+...+L_{i_{k}}} Gr_{o}(n, k)) occur L_{i_{k+1}}, …, L_{i_{n}} as well. This can be seen from the expression we have got.

So the equivariant Euler class of the tangent space at a general fix point δ = L_{i_{1}} + … + L_{i_{k}} of the action is

e(T_{δ}Gr_{o}(n, k)) = ∏_{s<t≤k}(ε_{i_{t}}L_{i_{t}} + ε_{i_{s}}L_{i_{s}}) ∏_{s≤k<t}(ε_{i_{s}}L_{i_{s}} + L_{i_{t}})(ε_{i_{s}}L_{i_{s}} − L_{i_{t}}).

The Thom polynomial of the cone is the following:

Example 7.8.

T p(cGr_{o}(n, k) =

= A ∑_{\begin{matrix}I∈\binom{n}{k} \\ \{e_{1},... ,ε_{k}\}∈\{±1\}^{k} \end{matrix}} ((ε_{1}L_{i_{1}} + … + ε_{k}L_{i_{k}}) ∏_{s,t∈I}(ε_{t}L_{i_{t}} + ε_{s}L_{i_{s}}) ∏_{\begin{matrix}1≤s≤k \\ t∉I \end{matrix}}(ε_{i_{s}}L_{i_{s}} + L_{t})(ε_{i_{s}}L_{i_{s}} − L_{t}))^{−1} ,

where

A = ∏_{1≤i_{1}≤i_{2}≤...≤i_{k}≤n}(L_{i_{1}} + L_{i_{2}} + … + L_{i_{k}})

Example ?? tells us the Thom polynomials of minimal orbits corresponding to highest weights Γ_{ω_{k}} for k = 1, … n − 2. It remains to determine the corresponding Thom polynomials for the remaining two spinor representations: the minimal orbits are called spinor varieties. Here are the results: First we give a short description of the spin representations, since these can be not constructed from the standard representation with the usual methods like making tensor products, exterior powers, symmetric powers, kernels or co-kernels of homomorphisms. Let Q be a nondegenerate symmetric bilinear form on a vector space V^{m}, the Clifford algebra C = C(Q) = Cliff(V, Q) is an associative algebra with unit 1, which contains and is generated by V, with v · v = Q(v, v) · 1 for all v ∈ V. Equivalently, we have the equation v · w + w · v = 2Q(v, w) for all v, w ∈ V. If e_{1}, e_{2}, …, e_{m} form a basis for V, then the products e_{i_{1}}e_{i_{2}} ⋯ e_{i_{k}} with e_{∅} = 1 form a basis for C(Q) = Cliff(V, Q), so dimC(Q) = 2^{m}. Now the Clifford algebra has a natural Z_{2} grading: C = C^{even} ⊕ C^{odd}, where C^{even} is spanned by products of an even number of elements in V, and C^{odd} by products of an odd number, in particular C^{even} is a subalgebra of dimension 2^{m−1}. Since C(Q) is an associative algebra, it determines a Lie algebra, with bracket [a, b] = ab − ba. The spinor representations of 𝔰𝔬_{m}ℂ can be found in two steps:

(i) embedding 𝔰𝔬_{m}ℂ inside the Lie algebra of the even part C^{even}. This can be done easily, see [?].

(ii) identifying the Clifford algebra with one or two copies of matrix algebras. To do this let V = W + W′ where W and W′ are n-dimensional isotropic subspaces for Q. With our standard Q on V = ℂ^{2n} W can be the subspace spanned by the first n basis vectors. Now we have isomorphism of algebras C(Q) ≅ End(∧^{•}W), where ∧^{•}W = ∧^{0}W ⊕ … ⊕ ∧^{n}W. Now we have a decomposition ∧^{•}W = ∧^{even}W ⊕ ∧^{odd}W into the sum of even and odd exterior powers, and C(W)^{even} respects this splitting. By the isomorphism C(Q) ≅ End(∧^{•}W) we have

C(Q)^{even} ≅ End(∧^{even}W) ⊕ End(∧^{odd}W)

so

𝔰𝔬(Q) ⊂ C(Q)^{even} ≅ 𝔤𝔩(∧^{even}W) ⊕ 𝔤𝔩(∧^{odd}W),

and hence we have two representations of 𝔰𝔬(Q) = 𝔰𝔬_{2n}ℂ, which we denote by

S^{+} = ∧^{even}W and S^{−} = ∧^{odd}W.

Proposition 7.9. The representations S^{±} are the irreducible representations of 𝔰𝔬_{2n}ℂ, with highest weights α = \frac{1}{2}(L_{1} + … + L_{n}) and β = \frac{1}{2}(L_{1} + … + L_{n−1} − L_{n} (the half-spin representations), more precisely

S^{+} = Γ_{α} and S^{−} = Γ_{β} if n is even
S^{+} = Γ_{β} and S^{−} = Γ_{α} if n is odd.

The representation S = S^{+} ⊕ S^{−} = ∧^{•}W is called the spin representation.

It is can be shown easily that the natural basis vectors e_{I} = e_{i_{1}} ∧ … ∧ e_{i_{k}} for ∧^{.}W span weight spaces with weight \frac{1}{2}(∑_{i∈I} L_{i} − ∑_{j∉I} L_{j}). All such weights with given I mod 2 are congruent by the Weyl group, so each of S^{+} and S^{−} must be an irreducible representation with highest weights trivially α and β.

For the odd case, write V = W ⊕ W′ ⊕ U where W and W′ are n-dimensional isotropic subspaces, and U is a one-dimensional space perpendicular to them. For the standard Q these are spanned by the first and second n basis vectors, and the last one. Now this decomposition determines an isomorphism of algebras C(Q) ≅ End(∧^{•}W) ⊕ End(∧^{•}W′). The subalgebra C(Q)^{even} of C(Q) is mapped onto either of the factors by this isomorphism, so we have

C(Q)^{even} ≅ End(∧^{•}W),

and we get

𝔰𝔬_{2n+1}ℂ = 𝔰𝔬(Q) ⊂ C(Q)^{even} ≅ 𝔤𝔩(∧^{•}W).

Proposition 7.10. The representation S = ∧^{•}W is the irreducible representation of 𝔰𝔬_{2n+1}ℂ with highest weight α = \frac{1}{2}(L_{1} +…+L_{n}), this is the spinor representation of 𝔰𝔬_{2n+1}ℂ.

The simply-connected covering spaces of the orthogonal groups are called spin groups Spin(Q). We have a two-sheeted covering homomorphism of SO(Q)

ρ : Spin(Q) → SO(Q)

with kernel {1, −1}. The spin representations are representations of Spin(Q). In the odd case the closed orbit of the spin representation S is the orthogonal Grassmannian of n-dimensional isotropic subspaces of ℂ^{2n+1}, the corresponding projective variety has dimension \binom{n+1}{2} in ℙ^{N}, N = 2^{n} − 1, and called the spinor variety, denoted by S or the variety of pure spinors. For Spin_{2n}ℂ the orthogonal Grassmannian of n-dimensional isotropic subspaces has two components, and the first is the minimal orbit of S^{+}, the other is of S^{−}. The corresponding spinor varieties S_{n}^{+} and S_{n}^{−} have dimension \binom{n}{2} in ℙ^{2^{n−1}−1}.

Remark 7.11. The spinor variety for Spin_{2n−1}ℂ is isomorphic to each of the spinor varieties for Spin_{2n}ℂ. In fact they are projectively equivalent as subvarieties of projective space ℙ^{2^{n−1}−1}. So it is enough to determine the Thom polynomial in the even case.

Since the Lie algebra of Spin(Q) is the same as that of SO(Q), we can use the method described above with 𝔰𝔬(Q). This is an example, where this method, which uses only the structure of the Lie algebra is more effective then the other method we mentioned earlier: the method of restriction equations. For computing restriction equations it should be necessary to understand the action of the Lie group Spin(Q) on S^{+} and S^{−}, which is quite complicated, and we have not succeeded to calculate the Thom polynomial by this method.

The highest weights of the representation S^{+} are

\{\frac{1}{2}(ε_{1}L_{1} + … + ε_{n}L_{n}\}\begin{matrix}\{ε_{1},... ,ε_{n}\}∈\{−1,+1\}^{n} \\ #\{−1\}=even \end{matrix} ,

while those of S^{−} are

\{\frac{1}{2}(ε_{1}L_{1} + … + ε_{n}L_{n}\}\begin{matrix}\{ε_{1},... ,ε_{n}\}∈\{−1,+1\}^{n} \\ #\{−1\}=odd \end{matrix} .

moreover, as we have seen before, these are all the weights of these representations. So we get

Example 7.12.

T p(S_{n}^{+}) = ∏_{\begin{matrix}ε_{1},... ,ε_{n}∈\{−1,+1\}^{n} \\ #\{−1\}=even \end{matrix}}(ε_{1}L_{1} + … + ε_{n}L_{n})·

· ∑_{\begin{matrix}ε_{1},... ,ε_{n}∈\{−1,+1\}^{n} \\ #\{−1\}=even \end{matrix}} ((ε_{1}L_{1} + … + ε_{n}L_{n}) ∏_{i<j}(−ε_{i}L_{i} − ε_{j}L_{j}))^{−1}

T p(S_{n}^{−}) = ∏_{\begin{matrix}ε_{1},... ,ε_{n}∈\{−1,+1\}^{n} \\ #\{−1\}=odd \end{matrix}} (ε_{1}L_{1} + … + ε_{n}L_{n}) ·

· ∑_{\begin{matrix}ε_{1},... ,ε_{n}∈\{−1,+1\}^{n} \\ #\{−1\}=odd \end{matrix}} ((ε_{1}L_{1} + … + ε_{n}L_{n}) ∏_{i<j}(−ε_{i}L_{i} − ε_{j}L_{j}))^{−1}

Unfortunately even the fastest personal computer “dies” for n > 5. In the case n = 5 the spinor varieties are 10-dimensional complex varieties in the 15-dimensional spinor representations, so T p(S_{n}^{±}) ∈ H^{10}(BSO_{10}ℂ) = ℤ, which is generated by the Euler-class. So the Thom polynomial is some integer multiple of the Euler class, and some computer computation gives that it is itself the Euler class, so T p(S_{5}^{±}) = L_{1} · . . . · L_{5}. It would be necessary to simplify the expression on the right hand side above before we use computers. This is our next goal: to determine the polynomial form of the Thom polynomial for these minimal orbits, by simplifying the expression.

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