## 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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