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Closure of Stokes matrices I: caterpillar points and Alekseev-Meinrenken diffeomorphisms | Xiaomeng Xu
; | Date: |
16 Dec 2019 | Abstract: | The Riemann-Hilbert maps of certain meromorphic linear systems with
Poncar$
m Acute{e}$ rank $1$ are diffeomorphisms $
u_u$, parametrized by the
regular elements of a Cartan subalgebra $uinh_{
m reg}(mathbb{R})$ of
$frak u(n)$, from the space $Herm(n)$ of $n imes n$ Hermitian matrices to
the space $Herm^+(n)$ of $n imes n$ positive definite Hermitian matrices. In
this paper, we propose an extension of the family of Riemann-Hilbert maps from
$h_{
m reg}(mathbb{R})$ to its de Concini-Procesi wonderful compactification
$mathcal{M}(mathbb{R})$ via isomonodromy deformation. We then study the map
$
u_{rel}$ corresponding to a caterpillar point on $mathcal{M}(mathbb{R})$,
and prove that (up to a gauge transformation) it coincides with the
Alekseev-Meinrenken diffeomorphism $Gamma_{AM}$ from $Herm(n)$ to
$Herm^+(n)$, a map uniquely characterized by distinguished linear algebra
properties. We also discuss the applications of our results in Poisson geometry
and representation theory. | Source: | arXiv, 1912.7196 | Services: | Forum | Review | PDF | Favorites |
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