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The notion of extended geometrically finite representations introduced by Weisman generalizes Anosov representations by studying the convergence dynamics on group boundary extensions. We prove that divergent, extended geometrically finite representations can be interpreted as admitting dominated splitting over certain flow spaces. In particular, we provide an example of such representation constructed by Tholozan–Wang in the study of simple Anosov representations.

The full Hecke algebra of a p-adic group G is complicated, but at Iwahori level, it is the specialization of an affine Hecke algebra at v = q. Lusztig defined the asymptotic Hecke algebra, which is a “limit” of the affine Hecke algebra as v goes to infinity. Although the “limiting process” does not make sense for the full Hecke algebra, Braverman and Kazhdan were able to extend the asymptotic Hecke algebra to arbitrary level. I will explain a proof of a conjecture by Bezrukavnikov, Braverman, and Kazhdan that the cocenter of the Hecke algebra is isomorphic to the cocenter of the asymptotic Hecke algebra.