Borel–Weil theorem

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

:$Gamma\_\{\; ext\{hol(G!/!T,L\_\{wlambda\})simeq\; V(lambda)$

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$L\_\{wlambda\}$ is the line bundle determined by the 1-dimensional representation$xi\_\{wlambda\}colon\; T\; omathbb\{C\}$ of "T" given by $xi\_\{wlambda\}(e^H),=e^\{(wlambda)(H)\}$ for $Hinmathfrak\{t\}$, the Cartan subalgebra according to "T".

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 $gTmapsto\; gB$ is a diffeomorphism of manifolds

:$G/Tstackrel\{simeq\}\{longrightarrow\}\; G\_mathbb\{C\}/B,$

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

Holomorphic sections

The smooth sections in $Gamma(G/T,,L\_lambda)$ can be thought of as smooth functions $fcolon\; G\; omathbb\{C\}$ such that $f(gt)=xi\_lambda(b^\{-1\}),f(g)$ for all $gin\; G$ and $tin\; T$. The action of "G" on these sections is given by $g,f(g\text{'})=f(g^\{-1\}g\text{'})$ for $g,g\text{'}in\; G$.

Now, let $mathfrak\; n$ denote the nilpotent part of the Lie algebra of "B". Then $Blongrightarrow\{he^He^Xmid\; hin\; T,\; Hin\; imathfrak\; t,\; Xinmathfrak\; n\}$ is a diffeomorphism. Hence $xi\_lambda$ 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 $hin\; T$, $Xin\; imathfrak\; t$ and $Hinmathfrak\; n$.

Since $G/Tsimeq\; G\_mathbb\{C\}/B$ we can now interpret the smooth sections as smooth functions $fcolon\; G\_mathbb\{C\}\; omathbb\{C\}$ such that

:$(*)qquadqquad\; f(gb)=xi\_lambda^mathbb\{C\}(b^\{-1\})f(g)$

for all $gin\; G$ and $bin\; B$. Hence the holomorphic sections correspond to holomorphic maps $fcolon\; G\_mathbb\{C\}\; omathbb\{C\}$ with the property $(*)$.

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 $X^t=-X$. 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 $T=\{\; ext\{diag\}(h,-h)mid\; hin\; imathbb\; R\}$. One can show, that the set of dominant analytical integral weights is given by

:$A\_\{\; ext\{dom(T)=\{\; frac\; n2epsilonmid\; ninmathbb\; N\_0\},$

where $epsilon(\; ext\{diag\}(h,-h))=h$.The Borel-Weil theorem provides an "SU"(2)-equivariant isomorphism $V(frac\; n2epsilon)simeqGamma\_\; ext\{hol\}(SU(2)/T,L\_\{-\; frac\; n2epsilon\})$. We now investigate the latter space.

Choose the Borel group, "B", to be the group of upper triangular matriceswhere $a,pinmathbb\; C$ with "a" nonzero.

Assume $fcolon\; SL(2,mathbb\; C)\; omathbb\; C$ satisfies $f(gb)=xi\_\{\; frac\; n2epsilon\}^mathbb\; C(b)f(g)$ for all $gin\; SU(2)$ and $bin\; B$.

If , then $xi\_\{\; frac\; n2epsilon\}^mathbb\; C(b)=a^n$, and hence, $f(gb)=a^nf(g)$. For and "a" = 1, we get

:

Hence, ƒ only depends of $z\_1$ and $z\_3$. Now, for "p" = 0

:

That is, ƒ is homogenous of degree "n". Since ƒ is holomorphic it's a polynomial, and if follows that $V(\; frac\; n2epsilon)$ is the space of homogenous polynomials $pcolonmathbb\; C^2\; omathbb\; C$ of degree "n".

History

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

References

*citation
last = Serre | first = Jean-Pierre | authorlink = Jean-Pierre Serre
title = Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d'après Armand Borel et André Weil)
journal = Séminaire Bourbaki
volume = 2 | issue = 100 | page = 447–454 | publisher = Soc. Math. France | location = Paris | year = 1995. In French; translated title: “Linear representations and Kähler homogeneous spaces of compact Lie groups (after Armand Borel and André Weil.”

*citation
last = Tits | first = Jacques | authorlink = Jacques Tits
title = Sur certaines classes d'espaces homogènes de groupes de Lie
series = Acad. Roy. Belg. Cl. Sci. Mém. Coll.
volume = 29 | year = 1955 In French.

*citation
last = Sepanski | first = Mark R.
title = Compact Lie groups.
series = Graduate Texts in Mathematics | volume = 235
publisher = Springer | location = New York | year = 2007.

*citation
last = Knapp | first = Anthony W.
title = Representation theory of semisimple groups: An overview based on examples
series = Princeton Landmarks in Mathematics
publisher = Princeton University Press | location = Princeton, NJ | year = 2001. Reprint of the 1986 original.