Geometry and topology of the tempered Iwahori-spherical representations of a split semisimple $p$-adic group

Let $\mathfrak{G}$ be a connected split $p$-adic group of type $B_n$, $C_n$ or $D_n$. Amongst the tempered representations of $\mathfrak{G}$ a key role is played by the Iwahori-spherical block. We provide a compact Hausdorff model for this space which allows us to compute the $K$-theory ranks for the corresponding $C^*$-algebra. The model, an extended quotient, is stratified by sectors. Underlying the approach of this paper is the interplay between the geometric extended quotient and the spectral extended quotient in the context of the ABPS conjecture.
We give geometric descriptions of every sector thus equipping the spectrum with a cellular structure. We classify the geometric structures arising in our model, and specifically discover real projective spaces along with cones and suspensions of these. These examples require a minor modification to our homotopy sector conjecture: we show that Langlands dual sectors are rationally (indeed dyadically) homotopy equivalent in all cases ($A_n, B_n, C_n, D_n, E_6, E_7$ and $E_8$).