University of Oford

We consider a discrete valuation ring R with field of fraction K and residue field k and a group scheme G connected, simply connected, split semisimple, affine algebraic group scheme over R with Lie algebra g_R. One defines the affinoid enveloping algebra to be the inverse limit of the standard enveloping algebra with respect to the \pi-adic filtration tensored with K. One would like a classification of the primitive spectrum of this ring. In this talk, I will define the affinoid Verma modules and show that they are "controlled" by the standard Verma modules. I will also explain the main difficulty of extending Dufflo's theorem which classifies the primitive spectrum of the standard enveloping algebra.