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  "body": "\n## Statement\n\nLet $G$ be a connected and simply connected complex [[semisimple Lie group|XJS07AJ50Y]], and let $T \\subseteq G$ be a maximal torus.\nLet $\\Phi$ be the roots of $G$ with respect to $T$, with a set $\\Phi^+$ of positive roots, and let $\\rho = \\frac{1}{2}\\sum_{\\alpha \\in \\Phi^+}\\alpha$.\nLet $B \\supseteq T$ be the [[Borel subgroup|5JSYA1JNCC]] whose Lie algebra is spanned by the Lie algebra of $T$ and the root spaces of the negative roots.\nLet $W$ be the [[Weyl group|EZGTEK3HQN]] with length function $\\ell$, and write $w \\cdot \\mu = w(\\mu + \\rho) - \\rho$.\nLet $\\mu$ be a character of $T$, extended to $B$.\nLet $L_\\mu$ be the holomorphic line bundle on the [[flag variety|NHRWCQQKTH]] $G/B$ defined as follows.\nIts sections over an open set $U \\subseteq G/B$ are the holomorphic $f$ on the preimage of $U$ in $G$ with $f(gb) = \\mu(b)^{-1} f(g)$ for $b \\in B$.\nThe group $G$ acts on the [[sheaf cohomology|PQ6Q143ARJ]] groups $H^i(G/B, L_\\mu)$ by left translation.\n\n1. If $\\mu$ is dominant, then $H^0(G/B, L_\\mu)$ is the [[irreducible|FX3NEFYFFN]] representation of highest weight $\\mu$, and $H^i(G/B, L_\\mu) = 0$ for $i \u003e 0$.\n2. If $\\mu + \\rho$ is orthogonal to some root, then $H^i(G/B, L_\\mu) = 0$ for all $i$.\n3. Otherwise there is a unique $w \\in W$ such that $\\lambda = w \\cdot \\mu$ is dominant; then $H^{\\ell(w)}(G/B, L_\\mu)$ is the irreducible representation of highest weight $\\lambda$, and $H^i(G/B, L_\\mu) = 0$ for $i \\ne \\ell(w)$.\n\n## Remarks\n\nPart 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.\nA 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 $s$.\nThis reduces the computation to the case of $SL(2,\\C)$.\nExample: for $G = SL(2, \\C)$, $G/B = \\mathbb{P}^1$ and $L_n \\cong \\mathcal{O}(n)$ for $n \\in \\Z$, with $\\rho = 1$.\nFor $n \\ge 0$, $H^0(\\mathbb{P}^1, \\mathcal{O}(n))$ is the space of homogeneous polynomials of degree $n$ in two variables, the irreducible representation of dimension $n + 1$.\nFor $n = -1$, $\\mu + \\rho = 0$, 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 $m + 1$.\nConventions 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.\nSince $G/B = K/T$ for a maximal compact subgroup $K$, the theorem also realises all irreducible representations of compact connected simply connected Lie groups.\n\n## Explanations\n\n### L1 human\n\nEvery finite-dimensional irreducible holomorphic representation of a simply connected complex semisimple group, or of a compact connected Lie group, can be built geometrically.\nOne takes the flag variety, a compact complex manifold attached to the group, and a line bundle on it chosen from a weight.\nThe holomorphic sections of the bundle form the representation with that highest weight.\nBott's part says what happens for the other weights: at most one cohomology group survives, and it is again irreducible.\n\n### L2 human\n\nThe theorem is a geometric companion of the [[Weyl character formula|230F5FXPMQ]]: applying the holomorphic Lefschetz fixed point formula to the action of $T$ on $G/B$ gives the character formula.\nIt is the model for many later constructions of representations from geometry, such as geometric representation theory and the Beilinson–Bernstein localisation.\n",
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