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A real analog of Kostant’s version of the Bott-Borel-Weil theorem. (English) Zbl 1090.17010

Let \({\mathfrak f}\) be a complex semisimple Lie algebra and fix a Cartan subalgebra \({\mathfrak h}\) and a corresponding triangular decomposition with set of positive (resp. negative) roots \(\triangle_+\) (resp. \(\triangle_-\)), defining in particular the Weyl group \(W\). The half sum of positive roots is denoted by \(R\). Fix furthermore a parabolic subalgebra \({\mathfrak q}\subset{\mathfrak f}\). We have a decomposition \({\mathfrak f}={\mathfrak f}_-\oplus{\mathfrak f}_0\oplus{\mathfrak q}_+\) with \({\mathfrak q}={\mathfrak f}_0\oplus{\mathfrak q}_+\) where as usually the reductive part \({\mathfrak f}_0\) includes the semisimple part of \({\mathfrak q}\) and the rest of \({\mathfrak h}\), while \({\mathfrak q}_+\) is the nilradical of \({\mathfrak q}\). As a third ingredient, let \(\lambda':{\mathfrak f}\to {\mathfrak g}{\mathfrak l}(V)\) be a finite dimensional representation of \({\mathfrak f}\) with highest weight \(\Lambda\), and consider the induced representation \[ \beta':{\mathfrak f}_0\to {\mathfrak g}{\mathfrak l}(H({\mathfrak q}_+;V)). \]
Then Kostant’s theorem [B. Kostant, Ann. Math. (2) 74, 329–387 (1961; Zbl 0134.03501)] states that the irreducible components of \(\beta'\) are in bijective correspondence with the set \(W^{\mathfrak q}\) of \(w\in W\) such that \(\Phi_w=w(\triangle_-)\cap\triangle_+\) contains only roots corresponding to \({\mathfrak q}_+\). Moreover, the highest weight of the irreducible component of \(\beta'\) corresponding to \(w\in W^{\mathfrak q}\) is \(w\cdot\Lambda=w(\Lambda+R)-R\) and it occurs in degree \(| w| \).
The investigation of the author concerns the analoguous situation for a real semisimple Lie algebra \({\mathfrak g}\) with complexification \({\mathfrak f}={\mathfrak g}({\mathbb C})\) and a parabolic subalgebra \({\mathfrak p}\subset{\mathfrak g}\) with associated decomposition \({\mathfrak g}={\mathfrak g}_-\oplus{\mathfrak g}_0\oplus{\mathfrak p}_+\). The main result is the description of the cohomology \(H({\mathfrak p}_+;V)\) using the representation \[ \beta:{\mathfrak g}_0\to {\mathfrak g}{\mathfrak l}(H({\mathfrak p}_+;V)). \] The anwer is given in terms of the Satake diagram of \({\mathfrak g}\), where one specifies \({\mathfrak p}\) by crossing out some vertices, the index of \(V\), the Hasse graph of \(H({\mathfrak p}_+;V)\) and other combinatorial data by using the relation to the known answer in the complex case.

MSC:

17B56 Cohomology of Lie (super)algebras
17B20 Simple, semisimple, reductive (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Citations:

Zbl 0134.03501
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