In the classification of unitary representations of the Lie group SL_2(R), the "discrete series" representations can be modeled on a space of L^2 functions on the upper half plane, and there's essentially one for each integer. What if R is replaced with the field Q_p of p-adic numbers? The group SL_2(Q_p) still has a discrete series, but it is much richer than that of SL_2(R). To study it, one can consider a p-adic version of the upper half plane. But strangely, the p-adic upper half plane isn't simply connected, and to unlock the full discrete series, one needs to study some of its finite covering spaces (the Lubin-Tate tower).

And for p-adic groups other than SL_2(Q_p)? This talk will be an invitation to the chain of beautiful ideas leading from the Lubin-Tate tower to a much more general notion of "local Shimura varieties" which are implicated in the Langlands program for p-adic fields.