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WHAT ARE FLAG MANIFOLDS AND WHY

ARE THEY INTERESTING?

David Wansbrough

Introduction

When I say that I study harmonic maps between flag manifolds, people invariably ask \lq\lq What is a flag manifold?''. Before starting my PhD I had never heard of them either and was surprised to find them in so many different branches of mathematics: differential geometry, algebraic geometry, representation theory, harmonic analysis of Lie groups, Coxeter groups, twistor theory, and mathematical physics. This article gives a simple introduction to flag manifolds and a taste of some of their connections to these diverse areas of mathematics.

Flag manifolds have many useful properties: they are compact, complex, homogeneous manifolds, and the complex and algebraic structures are closely related. Compactness ensures the finite dimensionality of cohomology spaces and allows integration on the manifold. As homogeneous spaces, flag manifolds can be represented as coset spaces of Lie groups. The complex structure makes available powerful tools from complex analysis. In particular, some problems about harmonic maps can be solved using holomorphic maps - the basic idea of twistor theory (see Burstall [4]).

The Definition

A flag is a sequence of nested subspaces:
\{0\} = V_0 \subset V_a \subset V_b \subset \cdots \subset V_n = {\Bbb C}^n

where \dim V_i=i. For each increasing sequence of integers (a,b,\ldots,n), the set of all such flags is a manifold called the flag manifold {\Bbb F}_{a,b,\ldots,n} (there are other notations).

The flags \{0\} \subset V_1 \subset V_2 \subset \cdots \subset V_n= {\Bbb C}^n are sometimes called full flags and {\Bbb F}_{1,2,\ldots,n} the full flag manifold; it has complex dimension \frac{n(n-1)}{2}. If any integers are skipped then we have partial flags and a partial flag manifold .

  • eg. The simplest flag manifold is the 2-sphere: S^2 is isomorphic to the complex projective space {\Bbb C}{\Bbb P}^1 (the set of lines in {\Bbb C}^2) via (x,y,z)\mapsto[x+iy,1-z], z\neq 1. Thus S^2 \cong {\Bbb C}{\Bbb P}^1 \cong {\Bbb F}_{1,2}

  • eg. Complex projective space, the set of lines in {\Bbb C}^n, {\Bbb C}{\Bbb P}^{n-1} \cong {\Bbb F}_{1,n}

  • eg. Grassmannian manifolds, the set of k-planes in {\Bbb C}^n, G_k({\Bbb C}^n) \cong {\Bbb F}_{k,n}

    These spaces are interesting in differential geometry because they have complicated geometries and topologies. Grassmannians are used in the construction of universal bundles and splitting bundles (see Bott & Tu [3]).

    Compact Complex Homogeneous Spaces

    Since invertible matrices preserve subspace inclusions, the Lie group SL_n (n\times n complex matrices with determinant 1) acts on {\Bbb F}_{a,\ldots,n} by left multiplication:
    V_i \to g V_i

    Claim: SL_n is transitive on {\Bbb F}_{a,\ldots,n}, ie. \forall x,y \in {\Bbb F}_{a,b,\ldots n}, \exists g\in SL_n such that x=g y (any two flags are related by a group element).

    Proof: A basis \{v_j\} of {\Bbb C}^n gives rise to a flag in a natural way: let V_i= \text{span} \{v_1,\ldots,v_i \}. All flags can be generated this way by choosing a linearly independent set of vectors in V_b-V_a, V_c-V_b, etc. Since GL_n can take any basis to any other, it is transitive on bases and hence also on flag manifolds. But many different bases generate the same flag; rescaling one basis vector will not change the flag generated, and in this way all flags can be generated by matrices with determinant 1. Thus SL_n is transitive on flag manifolds.

    We can do even better. Given any flag, it is possible to choose an orthonormal basis which generates the flag. Since the group SU_n (unitary matrices) is transitive on orthonormal bases, it is also transitive on flag manifolds. This is important because SU_n is a compact group.

    If a Lie group G acts transitively on a manifold X, and H is the stabilizer of a point x, then there is a well-defined bijection
    G/H \to X gH \mapsto gx

    In the case of flag manifolds these result in diffeomorphisms:
    {\Bbb F} _{a,\ldots,n} \cong SL_n/P \cong SU_n/K

    Where P or K is the stabilizer of any point. {\Bbb F} _{a,\ldots,n} inherits a complex structure from SL_n and compactness from SU_n.

    Much information about flag manifolds comes from Lie theory: root decompositions and representation theory in particular (see Baston & Eastwood [1]). Conversely, the geometry of Lie groups can be studied using flag manifolds.

    Parabolic Subgroups and Notation

    The stabilizers of the above SL_n action are called \B{parabolic} subgroups. For a given flag manifold, the stabilisers of different points are all conjugate to each other. In addition, all parabolic subgroups contain a conjugate of the subgroup of upper triangular matrices. The standard parabolic subgroup is the stabilizer of the flag defined by the standard basis of {\Bbb C}^n:

    \midspace{3cm}

    Both the conjugacy class of a parabolic subgroup and the flag manifold it defines are determined by the sizes of the GL_i blocks of the standard parabolic. These can be fixed by identifying the entries on the subdiagonal that are next to corners of two blocks (the entries will always be zero).

    The subdiagonal has n-1 entries and can be represented by a diagram with n-1 nodes (thanks to Mike Eastwood for the tex code):

    Placing a cross at the appropriate positions gives a notation which characterizes the conjugacy class of parabolics with representative P, and the flag manifold SL_n/P.

  • eg. S^2 = {\Bbb C} \times

  • eg. G_k({\Bbb C} ^n) =

  • eg. {\Bbb F} _{a,b,\ldots,m,n} = -1ex} \rightblob{ }

    Those familiar with the representation theory of Lie groups and algebras will recognise Dynkin diagrams and see that the notation really derives from the root space decomposition of SL_n. These ideas can be used to define generalised flag manifolds using Lie groups other than SL_n (see Baston & Eastwood [1]).

    Dynkin diagrams are a subset of the Coxeter graphs which arise from reflection groups (See Hiller [5]). One of the triumphs of Lie theory is the use of simple reflection geometry to express the structure of semisimple Lie algebras; providing information about Lie groups and flag manifolds.

    Projective Algebraic Varieties

    The following construction realizes a flag manifold as a projective algebraic variety, it uses three standard results from the representations of Lie groups:

  • A weight corresponds to coefficients for the nodes of a Dynkin diagram;

  • An irreducible representation corresponds to a weight with non-negative, integral coefficients;

  • The highest weight space of an irreducible representation is one-dimensional.

    To start, take a flag manifold {\Bbb F} _{a,\ldots,n}\cong SL_n/P and write 0 above crossed nodes and 1 above uncrossed nodes in its Dynkin diagram notation.

  • eg.

    This corresponds to a non-negative, integral weight which determines an irreducible representation of SL_n; ie. a vector space W with an SL_n action. The highest weight space is a one-dimensional subspace of W and a point x in the projective space {\Bbb P} W. SL_n also acts on {\Bbb P} W and the stabiliser of x is exactly the subgroup P. So the SL_n-orbit of x is diffeomorphic to SL_n/P, realising the flag manifold as a closed orbit in projective space and hence a projective algebraic variety. These are basic objects in algebraic geometry.

    Bott-Borel-Weil Theorem

    What follows is an elegant example of the interactions between representation theory, complex analysis, and the underlying geometry of flag manifolds. The Borel-Weil theorem induces representations from a subgroup of SL_n to the whole group. In this case the subgroup is B, the upper triangular matrices. Holomorphic maps play an important role and the process is sometimes called holomorphic induction . There are two steps: construct a one-dimensional representation of B and then induce a representation on SL_n.

    To get a one-dimensional representation of B, start with a non-negative integral weight. This is actually a one-dimensional representation of the diagonal matrices in SL_n. It can be extended trivially to all of B since the strictly upper triangular matrices are nilpotent.

    The induction step involves looking for new representations in the vector space of functions SL_n\to{\Bbb C} which are equivariant on B (the technique comes from induction on finite groups). The Borel-Weil theorem says the subspace of holomorphic equivariant functions is irreducible and is the dual of the irreducible representation corresponding to the weight we started with.

    Theorem 1 (Borel-Weil) If $\lambda$ is a non-negative integral weight of $SL_n$ then $$ \Gamma(\lambda) := \left\: SL_n\to{\Bbb C} , \textholomorphic , f(gb) = \lambda^{-1}(b)f(g) \textor } g\in SL_n, b\in B \right\}
    is the irreducible representation with lowest weight -\lambda.}

    Bott extended this theorem to parabolic subgroups other than B. The language is more general: there are roots and Weyl groups from Lie theory, representations are expressed in terms of vector bundles, and function spaces are replaced by cohomology groups. \rho is the weight with coefficient 1 at every node (half the sum of the positive roots).

    Theorem 2 (Bott-Borel-Weil) Let $P$ be a parabolic subgroup of $SL_n$ and $\lambda$ an integral weight which is non-negative on uncrossed nodes of $P$. If $\lambda + \rho$ is perpendicular to any root then $$ H^q(SL_{n}/P,{\Cal{O}}(E_{-\lambda})) = 0 \qquad \forall q $$ Otherwise there is only one non-trivial cohomology group and it realises an irreducible representation. $$ H^{l(w)}(SL_n/P,{\Cal{O}}(E_{-\lambda})) = V_{-w.\lambda} $$

    Here w is the unique element of the Weyl group that makes w.\lambda=w(\lambda+\rho)-\rho a non-negative weight, l(w) is the length of w in terms of simple reflections, {\Cal{O}}(E_{-\lambda}) is the sheaf of germs of holomorphic sections of the irreducible representation of P with lowest weight -\lambda, and V_{-w.\lambda} is the irreducible representation of SL_n with lowest weight -w.\lambda. If \lambda is non-negative on all nodes of P then w trivial and this reduces to the Borel-Weil theorem.

    References

  • [1] Robert J Baston and Michael G Eastwood. The Penrose Transform: Its interaction with Representation Theory , Oxford Mathematical Monographs, Oxford University Press (1989).

  • [2] Raoull Bott. Homogeneous Vector Bundles , Ann. Math. 60 , (1957), 203-248.

  • [3] Raoul Bott and Loring W Tu. Differential Forms in Algebraic Topology , Graduate Texts in Mathematics 82 , Springer Verlag (1982).

  • [4] Francis E Burstall. Twistor Methods for Harmonic Maps , Lecture Notes in Mathematics 1263 , (1985), 55-96.

  • [5] H Hiller. Geometry of Coxeter Groups , Research Notes in Mathematics 54 , Pitman Advanced Publishing Program (1982).

  • [6] Anthony W Knapp. Lie Groups, Lie Algebras, and Cohomology , Mathematical Notes 34 , Princeton University Press (1988).

  • [7] Wallach. Harmonic Analysis on Homogeneous Spaces , Pure and Applied Mathematics 19 , Marcel Dekker (1973).


    Department of Mathematics
    School of Mathematical Sciences
    The Australian National University


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