# Rolf Källström: Decompositions of D-modules over finite maps and the correspondence with representations of finite groups

Time: Wed 2020-03-04 15.30 - 16.30

Lecturer: Rolf Källström, Högskolan i Gävle

Location:

### Abstract

I start with modules over the ring of differential operators $${\mathcal D}_Y$$ on a variety $$Y$$ (or systems of linear partial differential equations) and some standard operations on such with respect to a finite map $$\pi : X\to Y$$ of smooth complex varieties, which we may as well assume is Galois with group $$G$$. In particular, given a semisimple $${\mathcal D}_X$$-module $$N$$ its direct image $$\pi_+(N)$$ is a $${\mathcal D}_Y$$-module which again is semisimple, and its decomposition is controlled by the representations of $$G$$. This is an algebraically proven particular case of the pretty hard decomposition theorem for D-modules, stating that the assertion is valid whenever $$\pi$$ is projective, which at present requires analysis. The category of finite-dimensional representations $$V$$ of $$G$$ corresponds to a certain category of $${\mathcal D}_Y$$-modules (Picard-Vessiot). This correspondence is not explicit, so it is an interesting problem to for instance determine a presentation $$M={\mathcal D}_Y/J$$ for some left ideal $$J$$ that corresponds to $$V$$. I will discuss an approach using ideas that originate with Klein, who treated the related problem to determine when a given $${\mathcal D}_Y$$-module $$M$$ is étale trivial (isotrivial), so that its pull-back $$\pi^!(M)$$ is a trivial $${\mathcal D}_X$$-module for some finite $$\pi$$. The classical case studied by Klein (and Fuchs...) is when $$M$$ is of rank 2 and $$Y$$ the projective line, which in the case when $$M$$ is a connection outside three points corresponds to hypergeometric equations; here the étale trivial hypergeometric connections are classified by the small Schwarz list, which in turn is controlled by the symmetry groups of the platonic solids.

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