Borel–Weil theorem

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In mathematics in the field of representation theory of compact Lie groups, the Borel–Weil theorem provides a concrete model for the irreducible representations as holomorphic sections of certain complex line bundles. It can be considered as a special case of the Borel–Bott–Weil theorem.

Given a compact connected Lie group G and an irreducible representation V of G, according to the highest weight theorem, there is a dominant analytical integral weight, λ, which completely determines V. Hence it makes sense to write V = V(λ).

The Borel-Weil theorem states that if λ is a dominant integral weight, then there is an equivalence

\Gamma_{\text{hol}}(G\!/\!T,L_{w\lambda})\simeq V(\lambda)

as G-representations. Thus the irreducible representation determined by the dominant weight λ is the space of holomorphic sections on G/T. Here T is a maximal torus in G, w is the uniquely determined Weyl group element mapping a positive Weyl chamber to its negative and Lwλ is the line bundle determined by the 1-dimensional representation \xi_{w\lambda}\colon T\to\mathbb{C} of T given by \xi_{w\lambda}(e^H)\,=e^{(w\lambda)(H)} for H\in\mathfrak{t}, the Cartan subalgebra according to T.

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[edit] Complex structure

Using the Peter–Weyl theorem, any compact Lie group can be viewed as a subgroup of the unitary group U(n) for a suitable natural number n. Consequently G is a real linear Lie group. Now, let T be a maximal torus in G. Let \mathfrak g_\mathbb{C} be the complexification of the Lie algebra \mathfrak g of G and let G_\mathbb{C} be the uniquely determined connected complex Lie subgroup of the general linear group GL(n,\mathbb{C}) with Lie algebra \mathfrak g_\mathbb{C}. Given a Borel subgroup B in G_\mathbb{C} it can be shown that the map gT\mapsto gB is a diffeomorphism of manifolds

G/T\stackrel{\simeq}{\longrightarrow} G_\mathbb{C}/B,

and G_\mathbb{C}/B has a natural structure as a complex manifold inherited from the complex Lie group G_\mathbb{C}.

[edit] Holomorphic sections

The smooth sections in \Gamma(G/T,\,L_\lambda) can be thought of as smooth functions f\colon G\to\mathbb{C} such that f(gt)=\xi_\lambda(b^{-1})\,f(g) for all g\in G and t\in T. The action of G on these sections is given by g\,f(g')=f(g^{-1}g') for g,g'\in G.

Now, let \mathfrak n denote the nilpotent part of the Lie algebra of B. Then B\longrightarrow\{he^He^X\mid h\in T, H\in i\mathfrak t, X\in\mathfrak n\} is a diffeomorphism. Hence ξλ can be extended to a 1-dimensional representation \xi_\lambda^\mathbb{C} of B by

\xi_\lambda^\mathbb{C}(he^He^X)=\xi_\lambda(h)e^{\lambda(H)},

with h\in T, X\in i\mathfrak t and H\in\mathfrak n.

Since G/T\simeq G_\mathbb{C}/B we can now interpret the smooth sections as smooth functions f\colon G_\mathbb{C}\to\mathbb{C} such that

(*)\qquad\qquad f(gb)=\xi_\lambda^\mathbb{C}(b^{-1})f(g)

for all g\in G and b\in B. Hence the holomorphic sections correspond to holomorphic maps f\colon G_\mathbb{C}\to\mathbb{C} with the property ( * ).

[edit] Example

Consider G = SU(2), the special unitary group. The Lie algebra of SU(2) is \mathfrak{su}(2) consisting of all 2 by 2 complex matrices with zero trace satisfying Xt = − X. Then G_\mathbb C=SL(2,\mathbb C) and \mathfrak{su}(2)_\mathbb C=\mathfrak{sl}(2,\mathbb C). A maximal torus T is given by diagonal matrices, that is T=\{\text{diag}(h,-h)\mid h\in i\mathbb R\}. One can show, that the set of dominant analytical integral weights is given by

A_{\text{dom}}(T)=\{\tfrac n2\epsilon\mid n\in\mathbb N_0\},

where ε(diag(h, − h)) = h. The Borel-Weil theorem provides a bijective SU(2)-intertwining operator V(\frac n2\epsilon)\simeq\Gamma_\text{hol}(SU(2)/T,L_{-\tfrac n2\epsilon}). We now investigate the latter space.

Choose the Borel group, B, to be the group of upper triangular matrices \begin{pmatrix} a & p\\ 0 & a^{-1} \end{pmatrix} where a,p\in\mathbb C with a nonzero.

Assume f\colon SL(2,\mathbb C)\to\mathbb C satisfies f(gb)=\xi_{\tfrac n2\epsilon}^\mathbb C(b)f(g) for all g\in SU(2) and b\in B.

If b=\begin{pmatrix} a & p\\ 0 & a^{-1} \end{pmatrix}\in B, then \xi_{\tfrac n2\epsilon}^\mathbb C(b)=a^n, and hence, f(gb) = anf(g). For g=\begin{pmatrix} z_1 & z_2\\ z_3 & z_4 \end{pmatrix}\in SU(2) and a = 1, we get

f\begin{pmatrix} z_1 & pz_1 + z_2\\ z_3 & pz_3 + z_4 \end{pmatrix}= f\begin{pmatrix} z_1 & z_2\\ z_3 & z_4 \end{pmatrix}.

Hence, ƒ only depends of z1 and z3. Now, for p = 0

f\begin{pmatrix} az_1 & a^{-1}z_2\\ az_3 & a^{-1}z_4 \end{pmatrix} =a^n f\begin{pmatrix} z_1 & z_2\\ z_3 & z_4 \end{pmatrix}.

That is, ƒ is homogenous of degree n. Since ƒ is holomorphic it's a polynomial, and if follows that V(\tfrac n2\epsilon) is the space of homogenous polynomials p\colon\mathbb C^2\to\mathbb C of degree n.

[edit] History

The theorem dates back to the early 1950s and can be found in Serre (1995) and Tits (1955).

[edit] References

  • Serre, Jean-Pierre (1995), “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, vol. 29, Acad. Roy. Belg. Cl. Sci. Mém. Coll.  In French.
  • Sepanski, Mark R. (2007), Compact Lie groups., vol. 235, Graduate Texts in Mathematics, 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.