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Realising The Smooth Representations of GL(2,Zp)

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If you have a question about this talk, please contact Tom Adams.

The character table of GL(2,Fq), for a prime power q, was constructed over a century ago. Many of these characters were determined via the explicit construction of a corresponding representation, but purely character-theoretic techniques were first used to compute the so-called discrete series characters. It was not until the 1970s that Drinfeld was able to explicitly construct the corresponding discrete series representations via l-adic ├ętale cohomology groups. This work was later generalised by Deligne and Lusztig to all finite groups of Lie type, giving rise to Deligne-Lusztig theory.

In a similar vein, we would like to construct the representations affording the (smooth) characters of compact groups like GL(2,Zp), where Zp is the ring of p-adic integers. Deligne-Lusztig theory suggests hunting for these representations inside certain cohomology groups. In this talk, I will consider one such approach using a non-archimedean analogue of de Rham cohomology.

This talk is part of the Junior Algebra and Number Theory seminar series.

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