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The Chevalley involution and a duality of weight varieties. (English) Zbl 1093.14067

Let \(G\) be a semisimple complex Lie group with parabolic subgroup \(P\) and maximal compact subgroup \(T\) of a Cartan subgroup \(H\subset P\). Given a suitable \(H\) linearized line bundle on the flag variety \(M=G/P\), one may form the Mumford quotient \(M//H\). Now let \(\theta\) be a Chevalley involution with \(\theta(t)=t^{-1}\) for \(t\in H\). (It is unique up to conjugation by an element of \(H\).) Then \(\theta\) induces an isomorphism between \(M//H\) and \(M^{\text{opp}}//H\), where \(M^{\text{opp}}=\theta(G)/\theta(P)\), with an appropriate \(H\) linearized line bundle on it. One of the aims of this paper is to make the isomorphism explicit, even on the coordinate ring level. When \(G=\text{GL}_n\) definite results are obtained in terms of tableaux. In the general case there is a conjecture in terms of the Littelmann path model. The paper also treats a similar problem with symplectic quotients. Here one starts with a \(\theta\) invariant maximal compact subgroup \(K\), a subtorus \(S\) of \(T\) and an element \(\mathbf{r}\) of the moment polyhedron for the action of \(T\) on \(M=K/Z(S)\). This time \(\theta\) induces an isomorphism of Kähler manifolds \(M//_{\mathbf{r}}T\to M//_{-\mathbf{r}}T\). One wants to make it explicit. In particular one wants to know when it is a self-duality, and if so, whether it is trivial. A classification of self-dualities is given for all simple \(G\).

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
53D20 Momentum maps; symplectic reduction
20G05 Representation theory for linear algebraic groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
14L24 Geometric invariant theory
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