- Borel–Weil theorem
In
mathematics in the field ofrepresentation theory ofcompact Lie group s, the Borel–Weil theorem provides a concrete model for theirreducible representation s asholomorphic section s of certain complexline bundle s. It can be considered as a special case of theBorel–Bott–Weil theorem .Given a compact
connected Lie group "G" and an irreducible representation "V" of "G", according to thehighest weight theorem , there is a dominant analytical integralweight , "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 determinedWeyl group element mapping a positiveWeyl chamber to its negative andL_{wlambda} is the line bundle determined by the 1-dimensional representationxi_{wlambda}colon T omathbb{C} of "T" given by xi_{wlambda}(e^H),=e^{(wlambda)(H)} for Hinmathfrak{t}, theCartan subalgebra according to "T".Complex structure
Using the
Peter–Weyl theorem , any compact Lie group can be viewed as asubgroup of theunitary group "U(n)" for a suitablenatural number "n". Consequently "G" is areal linear Lie group . Now, let "T" be a maximal torus in "G". Let mathfrak g_mathbb{C} be thecomplexification of theLie algebra mathfrak g of "G" and let G_mathbb{C} be the uniquely determined connected complex Lie subgroup of thegeneral linear group GL(n,mathbb{C}) with Lie algebra mathfrak g_mathbb{C}. Given aBorel 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')=f(g^{-1}g') for g,g'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 matricesegin{pmatrix}a & p\0 & a^{-1}end{pmatrix}where 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 b=egin{pmatrix}a & p\0 & a^{-1}end{pmatrix}in B, then xi_{ frac n2epsilon}^mathbb C(b)=a^n, and hence, f(gb)=a^nf(g). For g=egin{pmatrix}z_1 & z_2\z_3 & z_4end{pmatrix}in SU(2) and "a" = 1, we get
:fegin{pmatrix}z_1 & pz_1 + z_2\z_3 & pz_3 + z_4end{pmatrix}=fegin{pmatrix}z_1 & z_2\z_3 & z_4end{pmatrix}.
Hence, ƒ only depends of z_1 and z_3. Now, for "p" = 0
:fegin{pmatrix}az_1 & a^{-1}z_2\az_3 & a^{-1}z_4end{pmatrix}=a^n fegin{pmatrix}z_1 & z_2\z_3 & z_4end{pmatrix}.
That is, ƒ is
homogenous of degree "n". Since ƒ is holomorphic it's apolynomial , 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.
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