Borel–Weil theorem

In mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups and irreducible holomorphic representations of complex semisimple Lie groups. These representations are realized in the spaces of global sections of holomorphic line bundles on the flag manifold of the group. Its generalization to higher cohomology spaces is called the Borel–Weil–Bott theorem.

Statement of the theorem

The theorem can be stated either for a complex semisimple Lie group $G$ or for its compact form $K$ . Let $G$ be a connected complex semisimple Lie group, $B$ a Borel subgroup of $G$ , and $X = G / B$ the flag variety. In this scenario, $X$ is a complex manifold and a nonsingular algebraic $G$ -variety. The flag variety can also be described as a compact homogeneous space $K / T$ , where $T = K \cap B$ is a (compact) Cartan subgroup of $K$ . An integral weight $λ$ determines a $G$ -equivariant holomorphic line bundle $L λ$ on $X$ and the group $G$ acts on its space of global sections,

\Gamma (G/B,L_{\lambda }).\

The Borel–Weil theorem states that if $λ$ is a dominant integral weight then this representation is an holomorphic irreducible highest weight representation of $G$ with highest weight $λ$ . Its restriction to $K$ is an irreducible unitary representation of $K$ with highest weight $λ$ , and each irreducible unitary representations of $K$ is obtained in this way for a unique value of $λ$ . (A holomorphic representation of a complex Lie group is one for which the corresponding Lie algebra representation is complex linear.)

Concrete description

The weight $λ$ gives rise to a character (one-dimensional representation) of the Borel subgroup $B$ , which is denoted $χ λ$ . Holomorphic sections of the holomorphic line bundle $L λ$ over $G / B$ may be described more concretely as holomorphic maps

f:G\to {\mathbb {C}}_{{\lambda }}:f(gb)=\chi _{{\lambda }}(b)f(g)

for all $g \in G$ and $b \in B$ .

The action of $G$ on these sections is given by

g\cdot f(h)=f(g^{{-1}}h)

for $g, h \in G$ .

Example

Let $G$ be the complex special linear group $SL(2, C)$ , with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for $G$ may be identified with integers, with dominant weights corresponding to nonnegative integers, and the corresponding characters $χ n$ of $B$ have the form

\chi _{n}{\begin{pmatrix}a&b\\0&a^{{-1}}\end{pmatrix}}=a^{n}.

The flag variety $G / B$ may be identified with the complex projective line $CP 1$ with homogeneous coordinates $X, Y$ and the space of the global sections of the line bundle $L n$ is identified with the space of homogeneous polynomials of degree $n$ on $C 2$ . For $n \geq 0$ , this space has dimension $n + 1$ and forms an irreducible representation under the standard action of $G$ on the polynomial algebra $C [X, Y]$ . Weight vectors are given by monomials

X^{i}Y^{{n-i}},\quad 0\leq i\leq n

of weights $2 i - n$ , and the highest weight vector $X n$ has weight $n$ .

History

The theorem dates back to the early 1950s and can be found in Serre (1951-4) and Tits (1955).

References

Serre, Jean-Pierre (1954) [1951], "Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d'après Armand Borel et André Weil)", Séminaire Bourbaki, Paris: Soc. Math. France, 2 (100): 447–454 . In French; translated title: “Linear representations and Kähler homogeneous spaces of compact Lie groups (after Armand Borel and André Weil).”

Tits, Jacques (1955), Sur certaines classes d'espaces homogènes de groupes de Lie, Acad. Roy. Belg. Cl. Sci. Mém. Coll., 29 In French.

Sepanski, Mark R. (2007), Compact Lie groups., Graduate Texts in Mathematics, 235, New York: Springer .

Knapp, Anthony W. (2001), Representation theory of semisimple groups: An overview based on examples, Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press . Reprint of the 1986 original.

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