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- W2019446068 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 quan- tum levels for the case of the weight varieties associated to GLn(C). In par- ticular we obtain the following formula as a special case. Let r = (r1, . . . , rn) be an n-tuple of positive real numbers and Mr(CP m ) be the moduli space of semistable weighted (by r) configurations of n points in CP m modulo projec- tive equivalence, see for example (FM). Letbe the vector in R n with all components equal to P i ri/(m + 1). Then Mr(CP m ) � Mr(CP n m 2 ) (the meaning ofdepends on r and will be explained below, see Theorem 1.6). We conclude by studying 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 ob- tained as the monodromy of the Casimir connection, see (MTL), and will be treated in (HMTL)." @default.
- W2019446068 created "2016-06-24" @default.
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- W2019446068 date "2004-01-01" @default.
- W2019446068 modified "2023-09-25" @default.
- W2019446068 title "The Chevallry Involution and a Duality of Weight Varieties" @default.
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- W2019446068 doi "https://doi.org/10.4310/ajm.2004.v8.n4.a20" @default.
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