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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 Aminius 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 dobule 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.

3 added 50 characters in body

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 Heckewas 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 Aminius 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 dobule 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.

2 Part 2 of the answer

Essentially a

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 module bimodule of the Hecke algebra .and the parental group $G$.

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 Aminius 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 dobule 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.

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