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Analogues of Kostant’s \({\mathfrak u}\)-cohomology formulas for unitary highest weight modules. (English) Zbl 0651.17003
The author proves an analogue of a well-known theorem of Kostant: Theorem. Let (G,K) be a Hermitian symmetric pair with G classical and simple. Let X be an irreducible unitary highest weight module with highest weight \(\lambda -\rho.\) Then, for \(i\in {\mathbb{N}}\), \[ H^ i({\mathfrak u},X)\approx \oplus_{w}F({\mathfrak k},\bar w\lambda -\rho), \] where the sum is over \(w\in {\mathcal W}_{\lambda}^{{\mathfrak k},i}\). Notation. In addition to standard notation, \({\mathfrak u}\) is the abelian nilradical of a maximal parabolic subalgebra \({\mathfrak q}={\mathfrak k}\oplus {\mathfrak u}\) (complex Lie algebras). F(\({\mathfrak k},\lambda)\) is the irreducible finite-dimensional \({\mathfrak k}\)-module with highest weight \(\lambda\). \({\mathcal W}_{\lambda}\) is the subgroup of the Weyl group \({\mathcal W}\) generated by the reflections \(s_{\alpha}\) which satisfy: (i) \(\alpha\in \Delta (u)\) and (\(\lambda\),\({\check \alpha}\))\(\in {\mathbb{N}}^*\); (ii) if \(\beta\in \Delta\) and \((\lambda,\beta)=0\) then \((\alpha,\beta)=0\); (iii) id \(\beta\in \Delta\) is long and \((\lambda,\beta)=0\) then \(\alpha\) is short. Finally, \({\mathcal W}_{\lambda}^{{\mathfrak k},i}=\{w\in {\mathcal W}_{\lambda}|\) \(w\rho\) is \(\Delta^+_{\lambda}({\mathfrak k})\)-dominant and \(\ell (w)=i\}.\)
Corollary. Let \(D_ n=\prod_{\alpha \in \Delta (u)}(e^{\alpha /2}- e^{-\alpha /2}).\) Then \[ D_ n ch X=\sum_{w}(- 1)^{\ell_{\lambda}(w)} ch F({\mathfrak k},\bar w\lambda -\rho_{{\mathfrak k}}), \] summed over \(w\in {\mathcal W}^{{\mathfrak k}_{\lambda}}\).
The proof of the theorem relies, among other things, on the classification of unitary highest weight modules. The author mentions that the cohomology spaces in question were determined by Roger Howe for SU(p,q) and that the formulation given emerged in response to a question by him.
Reviewer: W.Rossmann

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
17B20 Simple, semisimple, reductive (super)algebras
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