Borel–Bott–Weil theorem

In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlier Borel–Weil theorem of Armand Borel and André Weil, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology.

Let G be a semisimple Lie group or algebraic group over ${\displaystyle \mathbb {C} }$, and fix a maximal torus T along with a Borel subgroup B which contains T. Let λ be an integral weight of T; λ defines in a natural way a one-dimensional representation C_λ of B, by pulling back the representation on T = B/U, where U is the unipotent radical of B. Since we can think of the projection map G → G/B as a principal B-bundle, for each C_λ we get an associated fiber bundle L_−λ on G/B (note the sign), which is obviously a line bundle. Identifying L_λ with its sheaf of holomorphic sections, we consider the sheaf cohomology groups ${\displaystyle H^{i}(G/B,\,L_{\lambda })}$. Since G acts on the total space of the bundle ${\displaystyle L_{\lambda }}$ by bundle automorphisms, this action naturally gives a G-module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups as G-modules.

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