(How) do Hecke-algebras arise naturally in your field of mathematics and why are they important?
How would you define them and how do you think about them?
e.g. generators and relations, functions on some space, grothendieck group…
[Math] Hecke-algebras in your field of mathematics
big-listbig-picturehecke-algebrassoft-question
Related Solutions
A Hecke algebra describes the most reasonable way to convolve functions or measures on a homogeneous space. Suppose that you have seen the definition of convolution of functions on a vector space, or on a discrete group --- the latter is just the group algebra of the group or some completion. Then how could you reasonably define convolution on a sphere? There is no rotationally invariant way to convolve a general $f$ with a general $g$. However, if $f$ is symmetric around a reference point, say the north pole, then you can define the convolution $f * g$, even if $g$ is arbitrary.
This is the basic idea of the Hecke algebra. The $(n-1)$-sphere is the homogeneous space $SO(n)/SO(n-1)$. A function $g$ on the sphere is a function on left cosets. A function $f$ on the sphere which is symmetric about a reference point is a function on double cosets. If $H \subseteq G$ is any pair of compact groups, if $f$ is any continuous function on $H\backslash G/H$, and if $g$ is any continuous function on $G/H$, then their product in the continuous group algebra is well-defined on $G/H$. The functions on double cosets make an algebra, the Hecke algebra, and the functions on left cosets are a bimodule of the Hecke algebra and the parental group $G$.
It is important for the same reasons that any other kind of convolution is important.
A particular case studied by Hecke Iwahori and others from before quantum algebra was the finite group $GL(n,q)$ and the upper triangular subgroup $B$. This is "the" Hecke algebra; it turns out that it is one algebra with a parameter $q$. Or as Ben says, this generalizes to the Iwahori-Hecke algebra of an algebraic group $G$ with a Borel subgroup $B$.
The other place that the Hecke algebra arises is as an interesting deformation of the symmetric group, or rather as a deformation of its group algebra. It has a parameter $q$ and you obtain the symmetric group when $q=1$. As I said, it is also the Hecke algebra of $GL(n,q)/GL(n,q)^+$, where $B = GL(n,q)^+$ is the Borel subgroup of upper-triangular matrices (all of them, not just the unipotent ones). There is a second motivation for the Hecke algebra that I should have mentioned: It immediately gives you a representation of the braid group, and this representation reasonably quickly leads to the Jones polynomial and even the HOMFLY polynomial.
When the Jones and HOMFLY were first discovered, it was simply a remark that the braid group representation was through the same Hecke algebra as the convolutional Hecke algebra for $GL(n,q)/B$ (or equivalently $SL(n,q)/B$). Even so, it's a really good question to confirm this "coincidence", as Arminius asks in the comment. Particularly because it is now a fundamental and useful relation and not a coincidence at all. As Ben explains in his blog post, the first model of the Hecke algebra is important for the categorification of the second model.
The coset space of $GL(n,q)/B$ consists of flags in $\mathbb{F}_q^n$, and you can see these more easily using projective geometry. When $n=2$, there is an identity double coset 1 and another double coset $T$. A flag is just a point in $\mathbb{P}^1$, and the action of $T$ is to replace the point by the formal sum of the other $q$ points. Thus you immediately get $T^2 = (q-1)T + q.$ When $n=3$, a flag is a point and a line containing it in $\mathbb{P}^2$. The two smallest double cosets other than the identity are $T_1$ and $T_2$. $T_1$ acts by moving the point in the line; $T_2$ acts by moving the line containing the point. A little geometry then gives you that $T_1T_2T_1$ and $T_2T_1T_2$ both yield one copy of the largest double coset and nothing else. Thus they are equal; this is the braid relation of the Hecke algebra. When $n \ge 3$, the Hecke algebra is given by these same local relations, which must still hold.
In general, if $A\subset B$ is a full reflective subcategory, then each object $a\in A$ is isomorphic to its image under the reflector. This seems to include many cases: $\mathbf{Ab}\subset \mathbf{Grp}$, $\mathbf{CompMet}\subset\mathbf{Met}$, $\mathbf{Top}_{n+1}\subseteq \mathbf{Top}_{n}$ (as in Why is Top_4 a reflective subcategory of Top_3?), etc.
EDIT: Following Pete L. Clark's comment, here is a clarification: The subcategory $A$ above is called reflective if the inclusion functor $A\subset B$ has a left adjoint, and full if this inclusion functor is full. In case $A$ is a reflective subcategory, the left adjoint to the inclusion functor is called a reflector.
Best Answer
Mine is pretty elementary:
Let $X$ be a $G$-set (for simplicity, let $X$ be a finite set, and $G$ be a finite group). Then the space $\mathbf C[X]$ of complex-valued functions on $X$ is a representation of $G$. The associated Hecke algebra is $\mathrm{End}_G \mathbf C[X]$.
For example, if $X$ is the variety of complete flags over a finite field, you end up with the usual finite dimensional Hecke algebra of type $A_n$. If $H$ is a subgroup of $G$ and $X=G/H$ then you end up with $\mathrm{End}_G \mathrm{Ind}_H^G 1$.
A basis of this Hecke algebra is indexed by relative positions of pairs of objects in $X$. What this means is that each linear endomorphism of $\mathbf C[X]$ is an integral operator $T_k$ with respect to some kernel $k:X\times X$ to $\mathbf C$. The kernels which give rise to $G$-endomorphisms are the ones which are constant on the orbits of the diagonal action of $G$ on $X\times X$ (which may be thought of as the set of relative positions of pairs in $X$).
I use these algebras to understand the permutation representations $\mathbf C[X]$. One may try, for example to compute the primitive central idempotents in this algebra. Often the set $X$ is a geometric object over a finite field of order $q$. In many such situations, $q$ enters the picture as a parameter which can then be replaced by an arbitrary complex number of a transcendental variable. Putting $q=1$ usually gives rise to some purely combinatorial object.