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In the following paper (in French), the authors consider the case of the $p$-adic upper half plane (the case of dimension $2$) and construct its coverings following Drinfeld. Boutot, J.-F. and Caray …

For discussions on forms of classical groups you can look at: -- André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. 24 (1961), 589-623 (also in Oeuvres Complètes). …

This is an old result due to Deligne: Le support du caractère d'une représentation supercuspidale. C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 4, Aii, A155–A157.

You may either work with an extended building as L. Spice suggests in his answer, or consider the following based on the fact that a generalized $BN$-pair contains a genuine $BN$-pair. With you nota …

You should find all ingredients needed for your calculation in the following very useful notes by Ralf Schmidt: Some remarks on local newforms for GL(2). J. Ramanujan Math. Soc. 17 (2002), 115-147 …

In fact $\phi$ is not only a (multiple of a) pseudo coefficient, but is a (multiple of a) coefficient of $\pi$. See e.g. Carayol's article "Représentations cuspidales du groupe linéaire", Ann. ENS. …

These Hecke algebras are intensively studied in the field of "type theory" for reductive $p$-adic groups. You have a nice summary of basic facts with proofs in chapter 4 of Bushnell and Kutzko's book …

Historically, the result is due to Hervé Jacquet MR0369624 (51 #5856) Jacquet, Hervé Sur les représentations des groupes réductifs p-adiques. C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Aii, A1271– …

The answer to both questions is yes. All irreducible supercuspidal representations of ${\rm GL}(N,F)$ are generic. See e.g. I. M. Gelfand and D. A. Kajdan, Representations of the group ${\rm GL}(n …

If you want a construction entirely compatible with Bushnell and Kutzko's theory of strata and simple characters (and that also works when $F$ has positive characteristic), you may refer to my PhD the …

computing the trace of a smooth irreducible representation is a very difficult problem which is far from being totally solved. For a nice overview, you may read : Sally, Paul J., Jr.; Spice, Loren C …

Existence of pseudo-coefficients for square-integrable representations (and the link with character values of the representations) is stated and proved in D. Kazhdan, Cuspidal geometry of $p$-adic g …

The general setting for your question is the theory of types as developped by Bushnell and Kutzko: Smooth representations of reductive p-adic groups: structure theory via types. Proc. London Math. So …

A reference for the result you are interested in is : Choucroun, F. M. (1996). Sous-groupes discrets des groupes p-adiques de rang un et arbres de Bruhat-Tits. Israel Journal of Mathematics, 93, 195- …

A reference for the "pro-unipotent radical" of a parahoric subgroup is given in Moy, Allen; Prasad, Gopal; (1994). "Unrefined minimal K-types for p -adic groups." Inventiones Mathematicae 116(1): 393 …