Kostant theorem on Lie algebra cohomology of nilpotent subalgebra
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introduction
- Humphreys 6.6
- At the end of his influential 1957 paper on the cohomology of vector bundles on homogeneous spaces such as flag varieties of semisimple Lie groups, Bott [43, §15] obtained what he described as a “curious corollary”: an explicit formula in terms of \(W\) for the dimensions of certain Lie algebra cohomology groups.
- He also remarked that this formula can be shown “without much trouble” to imply Weyl’s character formula.
- Although he worked in the setting of compact Lie groups and their complexifications, the essential point of the corollary is to describe the cohomology of a maximal nilpotent subalgebra of \(\mathfrak{g}\) (conjugate under the adjoint group to \(\mathfrak{n}\) or \(\mathfrak{n}^-\)) with coefficients in a finite dimensional simple module \(L(\lambda)\).
- Kostant [197] later developed these ideas further in the setting of Lie algebra cohomology.
- one can use the BGG resolution and the fact that for Verma modules \(H^i(\mathfrak{g},M(\mu))\) is \(\mathbb{C}_{\mu}\) for \(i=0\) for \(i>0\).
- this requires knowing the BGG resolution, which is a stronger result since it carries information about homomorphisms between Verma modules
- thm (Kostant)
Let \(\lambda\in \Lambda^{+}\). For a finite dimensional highest weight representation \(L({\lambda})\) of a complex semi-simple Lie algebra \(\mathfrak{g}\) \[ H^k(\mathfrak{n}^{-},L({\lambda}))=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} \]