Linear differential equations on $\mathbb{P}^{1}$ and root systems

Details Linear differential equations on $\mathbb{P}^{1}$ and root systems Linear differential equations on $\mathbb{P}^{1}$ and root systems

2010-10-13

arxiv.org/abs/1010.2580v4

Mathematics

Classical Analysis and ODEs

Scientific paper

We consider a linear differential operators on $\mathbb{P}^{1}$ having unramified irregular singular points. For this operator, we attach the root lattice of a Kac-Moody Lie algebra and the certain element in this lattice. Then we study the Euler transform for this differential operator and show that this translation by the Euler transform can be understand as the Weyl group action on the root lattice. Moreover we show that if the differential operator is irreducible, then the corresponding element becomes a root of this root system.

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