Borel–Weil–Bott theorem

Remarks

Part 1 is the Borel–Weil theorem, and parts 2 and 3 are Bott's extension; part 1 is the case $w = 1$ of part 3. A short proof due to Demazure uses fibrations of $G/B$ with fibre $\mathbb{P}^1$ to compare $L_\mu$ with $L_{s \cdot \mu}$ for simple reflections $s$. This reduces the computation to the case of $SL(2,\C)$. Example: for $G = SL(2, \C)$, $G/B = \mathbb{P}^1$ and $L_n \cong \mathcal{O}(n)$ for $n \in \Z$, with $\rho = 1$. For $n \ge 0$, $H^0(\mathbb{P}^1, \mathcal{O}(n))$ is the space of homogeneous polynomials of degree $n$ in two variables, the irreducible representation of dimension $n + 1$. For $n = -1$, $\mu + \rho = 0$, and indeed all cohomology of $\mathcal{O}(-1)$ vanishes; for $n = -m - 2$ with $m \ge 0$, $w$ is the reflection, $w \cdot \mu = m$, and $H^1(\mathbb{P}^1, \mathcal{O}(-m-2))$ has dimension $m + 1$. Conventions differ: with $B$ built from the positive roots, or with the opposite sign in $f(gb) = \mu(b)^{-1}f(g)$, one obtains dual representations instead. Since $G/B = K/T$ for a maximal compact subgroup $K$, the theorem also realises all irreducible representations of compact connected simply connected Lie groups.

Going deeper

For a student

The theorem is a geometric companion of the Weyl character formula: applying the holomorphic Lefschetz fixed point formula to the action of $T$ on $G/B$ gives the character formula. It is the model for many later constructions of representations from geometry, such as geometric representation theory and the Beilinson–Bernstein localisation.