TITLE: Relation between representations of p-adic groups and affine Hecke algebras
QUESTION [8 upvotes]: Let $R_n$ be the category of complex-valued smooth finite-length representations of the group $GL_n(F)$, where $F$ is a local field.
By the result of Borel, the subcategory of $R_n$ consisting of representations
that admit a Iwahori-invariant vector is equivalent to the category of finite-dimensional
representations of the affine Hecke algebra $H(n, F)$ attached to $GL_n(F)$. The equivalence translates parabolic induction to
module induction from $H(n_1, F) \otimes H(n_2, F)$ to $H(n_1 + n_2, F)$. 
This also follows from the results of the paper and the paper of Bushnell and Kutzko by type theory and the results by Heiermann.
Therefore to study the subcategory of $R_n$ consisting of representations
that admit a Iwahori-invariant vector is equivalent to study finite dimensional representations of affine Hecke algebras. This is very useful since we can apply results of representations of affine Hecke algebras to representations of complex-valued smooth finite-length representations of the group $GL_n(F)$.
Are there some references about similar results of the relation between classical groups of other types and some algebras similar to affine Hecke algebras.
Thank you very much.

REPLY [3 votes]: For the inner forms of $\mathrm{SL}_n$ it was done by Aubert, Baum, Plymen and Solleveld in "Hecke algebras for inner forms of p-adic special linear groups". Moreover, the equivalence between the category of representations in one Bernstein block and the category of modules over an affine Hecke algebra (may be extended by a finite group in the case of classical group), is true for all Bernstein blocks, not only for the Iwahori block.