Relation between representations of p-adic groups and affine Hecke algebras

Question

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.

Answer 1

Quite a lot is known beyond $GL_n$, but the results aren't always as clean and complete as for $GL_n$. This is (at least partly) because the Levi subgroups of more general reductive groups don't always look as "similar" to the full group as they do for $GL_n$ -- i.e. in $GL_n$ all Levi subgroups are products of smaller $GL_n$s, while in general you get Levi subgroups which are isomorphic to products of groups of various different types.

There are two things which you would need to generalize in order to get what you want. The first is a construction of a type for each block $R_s$ of the decomposition of the category $R(G)$ of smooth representations of $G$. Once you have a type $(J,\lambda)$ for $R_s$, you get an equivalence of categories between $R_s$ and modules over the $\lambda$-spherical Hecke-algebra $\mathcal{H}(G,\lambda)$. There are constructions of types for lots of $G$ and $s$ now: all blocks of $GL_n$ and its inner forms, all blocks of $SL_n$, and all blocks of classical groups (points in $GL_n(E)$ which are fixed by some Galois involution, for $E/F$ quadratic or trivial; essentially this means special orthogonal, symplectic or unitary) when the residue characteristic is odd. It seems like the construction of types for inner forms of classical groups isn't all that far away either, although it's certainly not yet complete. All of these constructions go through the same basic approach as the Bushnell--Kutzko theory; they're completely explicit and lead to types for which you'd hope the Hecke algebras are computable. There are probably a few other situations where something is known (spin groups, $G_2$), but I'm not so certain about what exactly we do know. There are also some results on the corresponding question for mod-$\ell$ representations of $GL_n$ and classical groups, but the relationship with Hecke algebras becomes much more complicated here.

The second is that, once you have a Hecke algebra $\mathcal{H}(G,\lambda)$ associated to a type, whether or not you can show that this is isomorphic to some tensor product of affine Hecke algebras, or something similar. Beyond $GL_n$, this is known for inner forms of $GL_n$, and for the blocks in classical groups which correspond to either the classical group itself, or a maximal proper Levi subgroup. I think there are some additional results in certain cases, such as for blocks defined on the Levi subgroup of a symplectic group which is of the form $GL_r^k\times Sp_{2n}$, but I'm not entirely sure which ones have been completely worked out. One notable omission from this list is $SL_n$; I don't think this has been written down anywhere, but it's understood since $SL_n$ is so nicely related to $GL_n$. The problem is that writing things down is a bit of a pain, because the Levi subgroups of $SL_n$ don't look very nice.

All of the above is via Bushnell--Kutzko theory. There are also completely general results on depth zero representations of arbitrary groups, which are quite a bit more straightforward.

The references for this are over a ton of papers, so instead of listing them I'll just point you to the various people in each case:

• $GL_n$ is due to Bushnell and Kutzko, as you know.
• Inner forms of $GL_n$ are due to Vincent Secherre and Shaun Stevens.
• $SL_n$ is Bushnell and Kutzko for the supercuspidal representations, and to David Goldberg and Alan Roche for the other blocks.
• Supercuspidal representations of classical groups are Shaun Stevens; types for the other blocks are Stevens and Michitaka Miyauchi. Stevens--Miyauchi also has the results on maximal Levis.
• Depth zero representations are due to Lawrence Morris.

Answer 2

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.