Stokes Matrices and Poisson Lie Groups

Details Stokes Matrices and Poisson Lie Groups Stokes Matrices and Poisson Lie Groups

2000-11-09

arxiv.org/abs/math/0011062v2

Invent. math. 146, 479-506 (2001)

Mathematics

Differential Geometry

23 pages, 1 figure, (top margin adjusted)

Scientific paper

We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc having an irregular singularity at the origin. The Riemann-Hilbert map for such connections, taking the Stokes data, induces a holomorphic map from the dual of the Lie algebra of G to the Poisson Lie group G*. The main result is that this map is Poisson. First this leads to new, more direct, proofs of theorems of Duistermaat and Ginzburg-Weinstein (enabling one to reduce Kostant's non-linear convexity theorem, involving the Iwasawa projection, to the linear convexity theorem, involving the `diagonal part'). Secondly we obtain a new approach to the braid group invariant Poisson structure on Dubrovin's local moduli space of semisimple Frobenius manifolds: it is induced from the standard Poisson structure on G*.

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