DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 303–308

THE DUISTERMAAT–HECKMAN INTEGRATION

FORMULA ON GENERALIZED FLAG MANIFOLDS

ANDREAS ARVANITOYEORGOS

Abstract. Let M be a 2n-dimensional symplectic manifold with symplectic 2-form ω. The partition function of a classical system whose dynamics is governed by the Hamiltonian function H can be written as an integral ∫_{M} e^{iϕH} \frac{ω^{n}}{n!} (ϕ ∈ ℝ). The Duistermaat-Heckman integration formula states that if the Hamiltonian vector field has only isolated zeros and it is almost periodic, then the previous integral can be expressed as a certain sum over the critical points of the Hamiltonian. We will study this integration formula for a class of Kähler manifolds, the generalized flag manifolds. These are homogeneous spaces of the form G/C(T), where G is a compact Lie group and T a maximal torus in G. Equivalently, they can be expressed as adjoint orbits of the adjoint representation of G in its Lie algebra. In our approach we use well known results of R. Bott and H. Samelson about the Morse theory of height functions on an adjoint orbit, to give a Lie algebraic expression for the Duistermaat-Heckman formula, suitable for calculations.

1. Introduction

Let M be a compact symplectic manifold of dimension 2n with symplectic 2form ω, and let T be a torus acting on M in a Hamiltonian way. This means that there is a momemt map H : M → 𝔱^{∗} (the Lie algebra of T), also called Hamiltonian, having the properties ι_{X}ω = −dH_{X} and ω(X, Y) = 0 for all X, Y ∈ 𝔱. Here ι_{X}ω is the contraction of the vector field X with the 1-form ω defined by ι_{X}ω(Y) = ω(X, Y) for all Y ∈ T M. The moment map is defined by ⟨X, H(m)⟩ = H_{X}(m), for all m ∈ M, X ∈ 𝔱, and assume in addition that for each X ∈ 𝔱 the infinitesimal action of (X^{∗}f)(m) = \frac{d}{dt}f(exp tX · m)|_{t=0} of X on M is given by the Hamiltonian vector field X̃ of the function H_{X}.

1991 Mathematics Subject Classification. Primary 53C80, 53C30, 53C21, 58F05; Secondary 57R70, 55N91.

Key words and phrases. Generalized flag manifolds, Duistermaat-Heckman formula, Morse theory, Homogeneous spaces, Equivariant cohomology.

The author was supported by a post-doctoral fellowship from the Greek State Scholarships Foundation (IKY), Programm #1 (1997-98) Grant #3.

This paper is in final form and no version of it will be submitted for publication elsewhere.

In the paper [Du-He] Duistermaat and Heckman showed that if a 2n-form μ on M is equivariantly closed, i.e. (d + ι_{X})μ = 0 and L_{X}μ = 0, then the value of the integral ∫_{M} μ is given in terms of contributions from the points in M which are fixed points of the vector field X.

Alternatively, their theorem assets that the push-forward H_{∗}(dm) of the symplectic (or Liouville) measure dm on M is a piecewise polynomial measure on 𝔱^{∗}. Furthermore, as shown in [At-Bo] the theorem of Duistermaat-Heckman can be viewed as a special case of more general results about equivariant cohomology (which uses the operator d + ι_{X} mentioned earlier, instead of d), but we will not discuss this here.

Now, when the vector field X has only isolated zeros or equivalently when H has only isolated critical points X_{cr}, then the above theorem leads to an exact formula for the oscillatory integral ∫_{M} exp(H)\frac{ω^{n}}{n!} given by

∫_{M} exp(H)\frac{ω^{n}}{n!} = ∑_{X_{cr}} \frac{exp(H(X_{cr}))}{D(X_{cr})}

where D(X_{cr}) are certain integers attached to the infinitesimal action of T at X_{cr}.

Recently, Picken [Pi] discussed the above integration formula on certain homogeneous spaces called generalized flag manifolds. These are spaces of the form G/C(T), where G is a compact Lie group and C(T) is the centralizer of a torus in G. Alternatively, these spaces are orbits under the adjoint representation of G in its Lie algebra 𝔤. The main attempt in that paper was to give a precise expression of the left-hand side of the above integration formula and of course to compute the right-hand side. The first goal was achived by using the Bruhat coordinates for generalized flag manifolds, but the second one was achived only for special examples.

In this paper we use well known results due to Bott and Samelson [Bo1], [Bo2], [Bo-Sa] about Morse theory for height functions on generalized flag manifolds to give an explicit general expression of the right-hand side of the Duistermaat-Heckman integration formula. It is given in terms of the roots of the Lie algebra of 𝔤 and it is suitable for computations.

For our purposes the Duistermaat-Heckman integration formula will have the form

(1)  ∫_{M} exp(H)\frac{ω^{n}}{n!} = ∑_{X_{cr}} \frac{e^{H(X_{cr})}}{\frac{det(Hess(X_{cr}))^{\frac{1}{2}}}{det(ω)}}

(see also [Pi] (5.8), (5.36)).

2. Generalized flag manifolds

We shall give a summary of the basic structure of the generalized flag manifolds or adjoint orbits. Let G be a semisimple and compact Lie group. A generalized flag manifold is a homogeneous space M = G/K whose isotropy subgroup K is the centralizer of a one-parameter subgroup T_{P} = \overline{exp ℝP} of G which generates a torus in G. Here P is an element in the Lie algebra 𝔤 of G. The homogeneous space M is also called adjoint orbit because in fact it can be identifed with the adjoint orbit Ad(G)P of P under the action of the adjoint representation Ad : G → Aut(𝔤). Without loss of generality the point P can be taken to lie in a fixed maximal torus T of G, which can also contain T_{P}. Hence we have that G/K = G/C(T_{P}) ≅ Ad(G)P.

Let G/K be a generalized flag manifold with 𝔤, 𝔨 the Lie algebras of G and K respectively. The Killing form B(X, Y) = tr ad(X) ∘ ad(Y) of 𝔤 gives rise to an orthogonal decomposition of 𝔤 as the direct sum 𝔤 = 𝔨 ⊕ 𝔪 with [𝔨, 𝔪] ⊂ 𝔪. The subspace 𝔪 is identified with the tangent space T_{0}M at the identity coset 0 = eK.

Let 𝔥 be the Lie algebra of T and 𝔥^{ℂ} its complexification. Then 𝔥^{ℂ} is a Cartan subalgebra of 𝔤^{ℂ}. Let R be the root system of 𝔤^{ℂ} with respect to 𝔥^{ℂ} and

𝔤^{ℂ} = 𝔥^{ℂ} + ∑_{α∈R} 𝔤^{(α)}

its root space decomposition. For any root α ∈ R we can choose elements E_{α} ∈ 𝔤^{(α)} which have the properties B(E_{α}, E_{−α}) = −1 and [E_{α}, E_{−α}] = −H_{α}, where H_{α} is determined by the equation B(H_{α}, H) = α(H) for all H ∈ 𝔥^{ℂ}. The corresponding real decomposition of 𝔤 may be written as

𝔤 = 𝔱 + ∑ V_{α}

where V_{α}^{ℂ} = ℂE_{α} ⊕ℂE_{−α}, and the sum is taken over one representative from each pair of roots ±α (there is no natural choice for each pair).

Since 𝔨^{ℂ} contains 𝔥^{ℂ}, there exists a subset R_{K} of R such that

𝔨^{ℂ} = 𝔥^{ℂ} + ∑_{α∈R_{K}} 𝔤^{(α)} .

The Lie algebra of the isotropy subgroup K = C(T_{P}) is just Ker(ad P), and the corresponding real decompotition is given by

𝔨 = 𝔱 + ∑_{α(P)=0} V_{α} .

Consequently, we obtain that 𝔪^{ℂ} = ∑_{α∈R_{M}} 𝔤^{(α)} and the root system R is decomposed as R = R_{K} ∪ R_{M}. The set R_{M} is called the set of complementary roots. There is a natural decomposition R_{M} = R_{M}^{+} ∪ R_{M}^{−}, where the positive complementary roots is the set R_{M}^{+} = {α ∈ R_{M} : α(P) > 0}. Hence we obtain that 𝔪 = ∑_{α∈R_{M}^{+}} V_{α}, and the set {E_{α} : α ∈ R_{M}} constitutes a basis for 𝔪^{ℂ}.

Next we recall the isotropy representation Ad^{G/K} of K on T_{0}M = 𝔪 which is the homomorphism defined by k ↦ (dL_{k})_{0} (here L_{g} is the diffeomorhism of G/K given by L_{g}(hK) = ghK for all g ∈ G). For a generalized flag manifold he isotropy representation is decomposed into a finite number of mutually inequivalent irreducible submodules, which can be described as follows. Let \overline{α}_{1}, …, \overline{α}_{k} be the distinct linear funcionals on 𝔱_{P} obtained by restricting the positive complementary roots to 𝔱_{P}. Let 𝔪_{i} = ∑_{α|_{𝔱_{P}} =\overline{α}_{i}} V_{α}. Then the required decomposition is

𝔪 = ∑_{i=1}^{k} 𝔪_{i} .

Finally, by using the embedding Ad(G)P ⊂ 𝔤, the tangent space to Ad(G)P at P is identified with Im(ad P), hence a typical point of the tangent space at P will have the form ad(P)X for some X ∈ 𝔤.

A generalized flag manifold acquires a natural G-invariant complex structure determined by the linear transformation J on 𝔪 given by

J = ∑_{α∈R_{M}^{+}} \frac{1}{2πα(P)} ad(P)|_{V_{α}} .

Notice that this complex structure depend on the choice of positive complementary roots and the choice of the point P.

Finally we will discuss G-invariant metrics on a generalized flag manifold M = G/K. These metrics are determined by ad(𝔨)-invariant inner products on Im(ad P). By Schur’s lemma any such inner product g_{P} can be expressed as

g_{P} = −c_{1} B|_{𝔪_{1}} − ⋯ − c_{k} B|_{𝔪_{k}} , (c_{i} > 0) .

A G-invariant metric of special importance is a Kähler metric g_{P}^{K} defined by g_{P}^{K}(ad(P)X, ad(P)Y) = ω(X, JY), where ω is the canonical symplectic (Kirillov) form determined by the skew symmetric billinear map Ω(ad(P)X, ad(P)Y) = −B(P, [X, Y]).

3. Critical points of height functions on generalized flag manifolds

For the following we refer to [Bo-Sa] and [Gu]. Let M = G/K be a generalized flag manifold viewed as an adjoint orbit Ad(G)P as above, and take a point Q ∈ 𝔤. We consider the height function

H^{Q}(X) = ⟨X, Q⟩ = −B(X, Q) .

Proposition 1. Let X be a point in Ad(G)P. Then X is a critical point of H^{Q} if and only if [Q, X] = 0.

Proof. The point X is a critical point of H^{Q} if and only if \frac{d}{dt}H^{Q}(Ad(exp tU)X)|_{t=0} = 0 for all U ∈ Im ad X, if and only if ⟨[U, X], Q⟩ = 0 for all U ∈ Im ad X, i.e [Q, X] = 0.

Corollary 1. The set of critical points of H^{Q} is Ad(G)P ∩ Ker(ad Q) or equivalentlly ⋃(Ad(G)P ∩ 𝔥), where the union is taken over all Cartan subalgebras 𝔥 that contain Q.

Without loss of generality we may assume that P is a critical point of H^{Q}, so [Q, P] = 0. Hence there exists a maximal torus whose Lie algebra contains both P and Q. Without loss of generality we may assume that T is such a maximal torus. We furthermore assume that Q is a regular point of 𝔱 which means that Ker ad(Q) is abelian, hence 𝔱 is the only Cartan subalgebra that contains Q. As a consequence of this the critical set is Ad(G)P ∩ 𝔱. Next we will compute the Hessian of H^{Q}.

Proposition 2. The Hessian of the function H^{Q} at a critical point X is given by

H(U, V) = ⟨ad(X) ad(Q)U, V ⟩

for all U, V ∈ Im ad(P).

Proof.

H(U, V) = \frac{d^{2}}{dsdt}H^{Q}(Ad(exp(sU + tV)X))|_{0} = ⟨ad(X) ad(Q)U, V ⟩

If we evaluate the Hessian at the basis {E_{α} : α ∈ R_{M}}, we obtain that (over ℂ) the set of eigenvalues of H^{Q} at a critical point X is {α(Q)α(X) : α ∈ R_{M}}.

4. The Duistermaat-Heckman integration formula on generalized flag manifolds

We are now in the position to give an expression of the right-hand side of the formula (1). We recall that the symplectic form ω is determined by the skew symmetric billinear map Ω(ad(P)X, ad(P)Y) = −B(P, [X, Y]). This can be evaluated by its restriction on each of the sets V_{α}, α ∈ R_{M}^{+}. By taking X = E_{α} and Y = E_{−α} we obtain that its complexification is given by Ω(ad(P)E_{α}, ad(P)E_{−α}) = α(P), hence det Ω = ∏_{α∈R_{M}^{+}} (α(P))^{2}. From the above analysis we obtain the following:

Theorem 1. The Duistermaat-Heckman integration formula (1) on a generalized flag manifold M = G/C(T_{P}) = Ad(G)P is given by

(2)  ∫_{M} exp(H^{Q})\frac{ω^{n}}{n!} = ( ∏_{α∈R_{M}^{+}} α(P)) ∑_{X_{cr}} \frac{e^{⟨X_{cr},Q⟩}}{∏_{α∈R_{M}} α(Q)α(X_{cr})}

where the sum is taken over the critical points X_{cr} of H^{Q}.

Example. Let M = SU(3)/T = SU(3)/S(U(1) × U(1) × U(1)) = Ad(SU(3))P, where P = diag(−i, 0, i) is an element in the Cartan subalgebra of the complexification of 𝔰𝔲(n), consisting of the set of matrices {diag(ix_{1}, ix_{2}, ix_{3}) : x_{1}+x_{2}+x_{3} = 0}. We fix an ordering so that the set of positive roots are R^{+} = {x_{1} − x_{2}, x_{1} − x_{3}, x_{2}−x_{3}}. We choose a regular point Q = diag(0, 2i, −2i) (i.e. (x_{i}−x_{j})(Q) ≠ 0), and let H^{Q}(X) = ⟨X, Q⟩ = −\frac{1}{6}B(X, Q) = − tr XY.

Without loss of generality the point P can be chosen to be a critical point of H^{Q}, say P = X_{1}, so the other critical points are X_{2} = diag(i, 0, i), X_{3} = diag(0, i, −i), X_{4} = diag(0, −i, i), X_{5} = diag(i, −i, 0), X_{6} = diag(−i, i, 0). (According to [Ra] the number of critical points for SU(n)/S(U(n_{1}) × ⋯ × U(n_{k})) is r = \frac{n!}{n_{1}!···n_{k}!} and in our case r = 6).

An easy computation shows that H^{Q}(X_{1}) = H^{Q}(X_{5}) = −2, H^{Q}(X_{2}) = H^{Q}(X_{6}) = 2, H^{Q}(X_{3}) = 4, H^{Q}(X_{4}) = −4. Furthermore, ∏_{α∈R^{+}} α(P) = 2i and ∏_{α∈R} α(Q)α(X_{i}) = 4^{5}, hence the right-hand side of (2) reduces to

\frac{2i}{4^{5}} [2e^{−2} + 2e^{2} + e^{4} + e^{−4}] .

References

[At-Bo] Atiyah, M. F., Bott, R., The moment map and equivariant cohomology, Topology 23

(1) (1984), 1–28.

[Bo1] Bott, R., Lectures on Morse Theory, Bonn (1958).

[Bo2] Bott, R., An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. de France 84 (1956), 251–282.

[Bo-Sa] Bott, R., Samelson, H., Applications of the theory of Morse to symmetric spaces, Amer.

J. Math. 80 (1958), 964–1029.

[Du-He] Duistermaat, J. J., Heckman, G. J., On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259–268. Addendum: 72 (1983), 153–158.

[Gu] Guest, M., Harmonic maps, Loop groups and Integrable Systems, Cambridge Univ. Press, LMS Student Texts 38 (1997).

[Pi] Picken, R. F., The Duistermaat-Heckman integration formula on flag manifolds, J. Math. Phys. 31 (3) (1990), 616–63.

[Ra] Ramanujam, S., Application of Morse theory to some homogeneous spaces, Tôhoku Math. J. 21 (1969), 343–353.

The American College of Greece (Deree) 6 Gravias Str., GR-15342 Aghia Paraskevi, GREECE E-email: taarv@hol.gr