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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

Location: Kräftriket, house 5, room 31

Participating: Rolf Källström, Högskolan i Gävle


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.