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Oct 18, 2016 at 12:41 comment added Paul Broussous I read the answers to the question. In fact it seems that Jianrong's question is only about the Iwahori-Spherical subcategory. In that restricted case a lot is known. By Borel one has been knowing since a long time ago that the trivial character of an Iwahori subgroup is a type in the sense of Bushnell and Kutzko for the Bernstein block of representations which are generated by their Iwahori spherical vectors (for any connected reductive $p$-adic group). Moreover the corresponding Hecke algebra is well known by the work of Iwahori-Matsumoto.
Oct 16, 2016 at 20:45 answer added kolmo timeline score: 3
Oct 16, 2016 at 13:00 comment added Amritanshu Prasad Is this paper: arxiv.org/abs/math/0504417 of any use?
Oct 16, 2016 at 12:59 answer added PL. timeline score: 10
Oct 16, 2016 at 9:28 history asked Jianrong Li CC BY-SA 3.0