Borel-Weil theorem

In mathematics, Borel-Weil's theorem gives a geometric description of the representations of Lie groups . It is a special case of the more general Borel-Weil-Bott theorem .

The theorem describes the representations of semisimple Lie groups; that is, it gives an explicit construction of the representations given by the theorem of highest weight .

construction

Let be a semi-simple Lie group , a Borel subgroup, and the flag variety . ${\displaystyle G}$ ${\displaystyle B\subset G}$ ${\displaystyle G/B}$

To a 1-dimensional representation one has a line bundle over defined by ${\displaystyle \lambda \colon B\to \mathbb {C} ^{*}}$ ${\displaystyle L_{\lambda }}$ ${\displaystyle G/B}$

${\displaystyle G\times \mathbb {C} /\sim }$

${\displaystyle (g,z)\sim (gb,\rho (b^{-1})z)\ \quad \forall b\in B}$

The effect of on , the holomorphic sections of this bundle of lines, defined by ${\displaystyle G}$ ${\displaystyle \Gamma (L_{\lambda })}$

${\displaystyle (g_{1}\cdot s)(g_{2}B)=s(g_{1}^{-1}g_{2}B)\ \quad \forall s\in \Gamma (L_{\lambda }),g_{1},g_{2}\in G}$

a representation of . ${\displaystyle G}$

Borel-Weil theorem

Let be a semisimple Lie group , a Borel subgroup and their decomposition as the product of their maximal torus and their unipotent radical . ${\displaystyle G}$ ${\displaystyle B\subset G}$ ${\displaystyle B=TU}$

If the constraint of to is a dominant integral element , then that representation of is whose greatest weight is the constraint of to . ${\displaystyle \lambda }$ ${\displaystyle T}$ ${\displaystyle \Gamma (L_{\lambda })}$ ${\displaystyle G}$ ${\displaystyle \lambda }$ ${\displaystyle T}$

Otherwise is . ${\displaystyle \Gamma (L_{\lambda })=0}$

Example: Representations of the SL (2, C)

For the Borel group we can choose the subgroup of the upper triangular matrices. Every 1-dimensional representation is of the form ${\displaystyle G=SL(2,C)}$ ${\displaystyle B}$

${\displaystyle \rho _{n}\left({\begin{array}{cc}a&b\\0&{\frac {1}{a}}\end{array}}\right)=a^{n}}$

for an integer . ${\displaystyle n}$

The Fahnenvarietät is with homogeneous coordinates and the sections of the line bundle for displaying correspond to the homogeneous polynomials of degree in the coordinates . These form an (n + 1) -dimensional vector space with a basis . So you get a representation of the dimension and rediscover the well-known proposition that there is an unambiguous, irreducible representation of for every dimension . ${\displaystyle G/B=\mathbb {C} P^{1}}$ ${\displaystyle X,Y}$ ${\displaystyle \rho _{n}}$ ${\displaystyle n}$ ${\displaystyle X,Y}$ ${\displaystyle X^{n},X^{n-1}Y,X^{n-2}Y^{2},\ldots ,Y^{n}}$ ${\displaystyle n+1}$ ${\displaystyle SL(2,\mathbb {C} )}$

Web links

• Jacob Lurie : A proof of the Borel-Weil-Bott theorem
• Borel-Weil Theorem (nLab)