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(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...

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They don't, yet. Perhaps you should delete the first word of your (interesting) question? – Yemon Choi Oct 11 '10 at 9:40
I put it in brackets as a compromise. If you don't like it feel free to edit my post and change it to what you see fit. – Jan Weidner Oct 11 '10 at 11:28
For some, the answer must be "What the heck are they?" Or we might find, like M. Jourdain, that we have been using them for years without knowing it. – Tom Goodwillie Oct 11 '10 at 18:38
You mean "What the hecke are they?" – Jan Weidner Oct 11 '10 at 19:40

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.

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One very interesting application is the proof by Kuhn and Priddy of the Whitehead Conjecture in stable homotopy theory (MR):

   author={Kuhn, Nicholas J.},
   author={Priddy, Stewart B.},
   title={The transfer and Whitehead's conjecture},
   journal={Math. Proc. Cambridge Philos. Soc.},
   review={\MR{803606 (87g:55030)}},

The point of contact is that the proof involves classifying spaces of elementary abelian groups $(\mathbb{Z}/p)^r$ and their suspension spectra. The $p$-local group ring $\mathbb{Z}_{(p)}[GL_r(\mathbb{Z}/p)]$ acts on this spectrum, and one can use the Steinberg idempotent to give a splitting. The relationship between the Steinberg idempotent and Hecke operators is pure algebra.

There is also a relationship between Hecke operators and power operations in elliptic cohomology (MR):

   author={Ando, Matthew},
   title={Power operations in elliptic cohomology and representations of
   loop groups},
   journal={Trans. Amer. Math. Soc.},
   review={\MR{1637129 (2001b:55016)}},

This actually generalises to give a relationship between power operations in Morava $E$-theory (at the prime $p$ and height $n$) and a kind of Hecke algebra for the monoid of $n\times n$ matrices over $\mathbb{Z}_p$ with nonzero determinant.

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Here is an interesting paper by Bost and Connes that combines Hecke algebras with type III factors (i.e., von Neumann algebras) and number theory:

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Mine is pretty standard. Denoting $\mathcal{U} = \mathcal{U}_q(\mathfrak{sl}_N)$ and $V$ the fundamental representation of $\mathcal{U}$, the braiding $\hat{R} : V \otimes V \to V \otimes V$ gives rise to a representation of the Hecke algebra $H_m(q)$ on $V^{\otimes m}$, which is the commutant of the representation of $\mathcal{U}$ on $V^{\otimes m}$. This gives a $q$-analogue of Schur-Weyl duality and tells you how to decompose $V^{\otimes m}$ as a direct sum of irreps of $\mathcal{U}$. For this perspective, one defines the Hecke algebra via generators and relations.

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