The Chevallry Involution and a Duality of Weight Varieties

Abstract

In this paper we show that the classical notion of association of projective point sets,[DO], Chapter III, is a special case of a general duality between weight varieties (i.e. torus quotients of flag manifolds) of a reductive group G induced by the action of the Chevalley involution on the set of these quotients. We compute the dualities explicitly on both the classical and quantum levels for the case of the weight varieties associated to GL_n(C). In particular we obtain the following formula as a special case. Let r = (r₁, ...,r_n)be an n-tuple of positive real numbers and M_r(CP^m)be the moduli space of semistable weighted(by r)configurations of n points in CP^m modulo projective equivalence, see for example[FM]. Let LAMBDA be the vector in Rⁿ with all components equal to [Sigma]_ir _i/(m + 1). Then M_r(CP ^m) is isomorphic to M_LAMBDA-rr (CP^n-m-2) (the meaning of is isomorphic to depends on r and will be explained below, see Theorem 1.6). We conclude by studying "self-duality" i.e. those cases where the duality isomorphism carries the torus quotient into itself. We characterize when such a self-duality is trivial, i.e. equal to the identity map. In particular we show that all self-dualities are nontrival for the weight varieties associated to the exceptional groups. The quantum version of this problem, i.e. determining for which self-adjoint representations V of G the Chevalley involution acts as a scalar on the zero weight space V[0], is important in connection with the irreducibility of the representations of Artin groups of Lie type which are obtained as the monodromy of the Casimir connection, see [MTL], and will be treated in [HMTL].