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We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid   G  ‐connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy–Prasad filtration associated to a point in the Bruhat–Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus   S  , then we have an   S  ‐toral connection. In this language, the irregular singularity of the Frenkel–Gross connection gives rise to the homogeneous toral connection of minimal slope associated to the Coxeter torus   C  . In the present paper, we consider connections on    G  m    which have an irregular homogeneous   C  ‐toral singularity at zero of slope    i  /  h   , where   h   is the Coxeter number and   i   is a positive integer coprime to   h  , and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.

Rigid connections on P 1 via the Bruhat–Tits building