# Connection between Hecke operators and Hecke algebras

Hecke operators are things that act on modular forms and give rise to a lot of interesting arithmetical results:

http://en.wikipedia.org/wiki/Hecke_operator

On the other hand on the wikpedia page for Hecke algebras, which should naively be an algebra of Hecke operators, the term seems to acquire very different meanings, such as deformations of group algebra of a Coxeter group.

What is the connection between the two?

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There are at least two MO questions about this: mathoverflow.net/questions/19684/… and mathoverflow.net/questions/4547/definitions-of-hecke-algebras . –  Qiaochu Yuan Nov 29 '10 at 0:51
The connection is also explained at the Wikipedia article on Hecke algebras: en.wikipedia.org/wiki/… –  Qiaochu Yuan Nov 29 '10 at 0:59

There are basically 3 senses of "Hecke algebra", and they are related to each other. The modular-form sense is a special case of all three.

The oldest version is that motivated by modular forms, if we think of modular forms as functions on (homothety classes of) lattices: the operator $T_p$ takes the average of a $\mathbb C$-valued function over lattices of index $p$ inside a given lattice. Viewing a point $z$ in the upper half-plane as giving the lattice $\mathbb Z z + \mathbb Z$ makes the connection to modular forms of a complex variable.

One important generalization of this idea is through repn theory, realizing that when modular forms are recast as functions on adele groups, the p-adic group $GL_2(\mathbb Q_p)$ acts on modular forms $f$. To say that $p$ does not divide the level becomes the assertion that $f$ is invariant under the (maximal) compact subgroup $GL_2(\mathbb Z_p)$ of $GL_2(\mathbb Q_p)$. Some "conversion" computations show that $T_p$ and its powers become integral operators (often mis-named "convolution operators"... despite several technical reasons not to call them this) of the form $f(g) \rightarrow \int_{GL_2(\mathbb Q_p)} \eta(h)\,f(gh)\,dh$, where $\eta$ is a left-and-right $GL_2(\mathbb Z_p)$-invariant compactly-supported function on $GL_2(\mathbb Q_p)$. The convolution algebra (yes!) of such functions $\eta$ is the (spherical) Hecke algebra on $GL_2(\mathbb Q_p)$.

A slightly larger, non-commutative convolution algebra of functions on $GL_2(\mathbb Q_p)$ consists of those left-and-right invariant by the Iwahori subgroup of matrices $\pmatrix{a & b \cr pc & d}$ in $GL_2(\mathbb Z_p)$, that is, where the lower left entry is divisible by $p$. This algebra of operators still has clear structure, with structure constants depending on the residue field cardinality, here just $p$. (The Iwahori subgroup corresponds to "level" divisible by $p$, but not by $p^2$.) This is the Hecke algebra attached to the affine Coxeter group $\hat{A}_1$.

Replacing $p$ by $q$, and letting it be a "variable" or "indeterminate" gives an example of another generalization of "Hecke algebra".

The latter situation also connects to "quantum" stuff, but I'm not competent to discuss that.

Edit: by now, there are several references for the relation between "classical Hecke operators" (on modular forms) and the group-theoretic, or representation-theoretic, version. Gelbart's 1974 book may have been the first generally-accessible source, though Gelfand-PiatetskiShapiro's 1964 book on automorphic forms certainly contains some form of this. Since that time, Dan Bump's book on automorphic forms certainly contains a discussion of the two notions, and transition between the two. My old book on Hilbert modular forms contains such a comparison, also, but the book is out of print and was created in a time prior to reasonable electronic files, unfortunately.

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I was under the impression that the last definition in Wikipedia ("the centralizer algebra of an induced representation") is the most general. Is that impression accurate? –  Qiaochu Yuan Jul 29 '11 at 16:33
I think for instances of the group-theoretic sense, maybe, but not the "q-" version, unless, conceivably, the "quantum" viewpoint includes that. For the Iwahori-Hecke case, the algebra itself is not commutative, and/but the "Bernstein center" is arguably better to consider? It's not clear to me how to unify the various notions, truly. The Kutzko-Bushnell construction of supercuspidals on $GL_n(\mathbb Q_p)$ shows that for a surprising variety of compact opens, the corresponding Iwahori-like-Hecke algebra is a "generalized" (non-comm) Hecke algebra. Maybe your interpretation is possible? –  paul garrett Jul 29 '11 at 16:50
What's a good reference for the representation theory aspects of Hecke theory that you describe? –  Jenifer Jul 3 '12 at 17:21