Hecke algebra of GL(2,F)

Question

I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to correct me in the probably various places I am wrong)

I am trying to make some basic computations, like for example compute the whole algebra probably, $H_K$ (the left-invariant on $K$ distributions) and the center (which in the rest of literature is what is usually denoted by $H_K$ I think, essentially the bi-$K$-invariant distributions). Now the center can be computed by the Satake isomorphism (and it is isomorphic to $\mathbb{C}[z_1^{\pm},z_2^{\pm}]^{S_2}$). For the rest I really could not come up with an explicit computation.

Now searching around, I found that most sources do not define the Hecke algebra as the locally constant compactly supported distributions, but by a purely algebraic definition with some generators over the Weyl group. I really cannot understand this definition.

Can you provide me with some source that explains this explicit presentation of the Hecke algebra, and why is it the same as Bernstein's one?

Answer 1

Let $G$ be a reductive $p$-adic group. First fixing a Haar measure on $G$, you can identify the algebra of distributions of $G$ with the ("big") Hecke algebra $H(G)$ of locally constant complex functions with compact support equipped with convolution $\star$. If $K$ is any compact open subgroup, the bi-$K$-invariant functions form a subalgebra $H(G,K)$, called the $K$-spherical Hecke algebra. More generally if $\rho$ is an irreducible smooth representation of $K$, you may form the subalgebra $H(G,\rho ) = e_\rho \star H(G)\star e_\rho$, where $e_\rho \in H(G)$ is the idempotent attached to $\rho$. Note that $H(G,K)$ corresponds to the particular case where $e_\rho$ is the trivial character.

All these algebras are called Hecke algebras. On the other hand there are standard Hecke algebras defined in an algebraic manner via generators and relations, like Iwahori-Hecke algebras.

You cannot expect to understand any sort of Hecke algebra in a naive approach. Any progress in the description of these algebras rely on a very fine understanding of the structure of $G$ (in particular of the double coset set $K\backslash G/K$) as well as a deep understanding of the structure of the representations. In general one wants to find an explicit isomorphism between an Hecke algebra that one wants to understand with a standard Hecke algebra.

Historically the first results concerned the spherical Hecke algebra $H(G,K)$, where $K$ is a "special" maximal subgroup of $G$ (because of its importance in the theory of automorphic forms) via the Satake isomorphism and the Iwahori Hecke algebra (when $K$ is an Iwahori subgroup), because of its importance in the study of unramified principal series.

The general understanding of the Hecke algebras that arise from reductive $p$-adic groups is one of the aims of Type Theory as conceptualized by Bushnell and Kutzko. It turns out that a lot of those Hecke algebras are in fact isomorphic to (or at least Morita equivalent to) standard Hecke algebras! But this is not trivial: this is the result of decades of research starting in the $50$'s.

Answer 2

A Hecke algebra, in the most common definition, is associated to a pair of groups $G$ and $K$, and is the convolution algebra of bi-$K$-invariant distributions on $G$.

As far as I know the left-invariant distributions are never considered to be a Hecke algebra. This algebra has a highly degenerate structure, because if $a,b$ are distributions and $a'$ is the average of the right translates of $a$ by elements of $K$, then $ab=a'b$, so $(a-a')b$ is zero. So the algebra has a huge space of zero-divisors, and in particular does not have a unit. If you mod out by the degenerate elements $a-a'$, you obtain the usual Hecke algebra of bi-invariant functions.

The definitions you see involving the Weyl group are most likely Iwahori-Hecke algebras, as Marty points out. In this case we would take, if $F =\mathbb Q_p$, $K$ to be the subgroup of elements of $GL_2(\mathbb Z_p)$ that are congruent to upper-triangular matrices mod $p$.

In addition to the link Marty provided, I think your idea of trying to work out at least some of the calculations yourself in the $GL_2$ case is a good one, by using the Bruhat decomposition to find $K \backslash GL_2 (F) / K$ and then computing the products of some double cosets.