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Several times I have come across these algebras and I wonder why any of these are interesting; I'm very sure they are, but I could not find an answer in the literature.

For example (the very general version of) a graded Hecke algebra for a finite group $G$ acting on a finite dimensional $\mathbb{C}$-vector space $V$ is defined as an algebra $A_\kappa := (T(V) \sharp \mathbb{C}G)/\langle vw - wv - \kappa(v,w) \mid v,w \in V \rangle$ satisfying the PBW-property $S(V) \sharp \mathbb{C}G \cong \mathrm{gr}(A_\kappa)$, where $\kappa:V \times V \rightarrow \mathbb{C}G$ is an alternating bilinear form. For these algebras I've seen an explanation like "we want to study the geometry of the action of $G$ on $V$, but the commutative algebraic geometry, i.e. $S(V)^G$, is bad, so we better study $S(V) \sharp \mathbb{C}G$"; but then I don't know why I'm interested in those $A_\kappa$. Can anybody explain this and make this precise?

I was also told that graded Hecke algebras (for Weyl groups) were introduced by Lusztig in order to study affine Hecke algebras which in turn are important in the representation theory of semisimple split $p$-adic groups. Do the general graded Hecke algebras above have a similar use?

The Cherednik and symplectic reflection algebras are special cases of graded Hecke algebras. If I would have a nice motivation for graded Hecke algebras, I would also have one for them; but I am pretty sure that there are different motivations for them!?

I am thankful for any explanations and hints to literature.

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In addition to many excellent answers posted so far, I would like to explain another way in which the relations of the "graded" or degenerate affine Hecke algebras arise in representation theory, which I find most helpful in understanding them (and is how I would recover them for myself if stranded on a desert island). This introduction has the advantage that it could be explained to an undergraduate who knew what a surface and a fundamental group was. All of this is in Cherednik's book in one form or another.

In this point of view, the "full" DAHA and AHA are the primary objects, and one recovers the degenerate versions by a process similar to the degeneration from $U_q(\mathfrak{g})$ to $U(\mathfrak{g})$. Let me just discuss $A_n$, although an analogous construction works for any Weyl group (and parts of the discussion for any complex reflection or symplectic reflection group).

The Artin braid group $B_n$ is $\pi_1(C_n(\mathbb{C}))$, the fundamental group of configurations of $n$ points in $\mathbb{C}$. It's easy to identify this with the braid group associated to the root system of type $A_{n-1}$, since the reflection hyperplanes are precisely what impose the distinctness of the points in the configuration. Denote the usual generators of $B_n$ by $T_i$.

One may instead consider the "double affine braid group" $DB_n$, $\pi_1(C_n(E))$, which is the configuration space of $n$ points on an elliptic curve (or what matters topologically is that it's a $S^1\times S^1$). One finds generators $B_n$ corresponding to loops which are contained in a contractible ball, and new generators $X_1,\ldots, X_n, Y_1,\ldots, Y_n$ corresponding to taking the $i$th point of the configuration and running it around the the inside our outside ring of the torus. One computes relations that the $X_i$ commute, the $Y_i$ commute, and $T_iX_iT_i=X_{i+1}$, $T_iY_{i+1}T_i=Y_i$, and (Y_2,X_1):=Y_2X_1Y_2^{-1}X_1^{-1}=T_1^{-2}$.

Now, how is this related to the algebras in your question? Let $\tilde{A}$ denote the group algebra of $DB_n$ with coefficients in $\mathbb{C}[[\hbar]]$, completed in the $\hbar$-adic topology. Let $A$ denote the quotient by the additional relations

$(T_i-q)(T_i+q^{-1})$, where $q:=e^{\hbar/2}$.

Now suppose that $V$ is some representation of $A$ such that $Y_i$ acts as $1$ modulo $\hbar$. In this case, it makes sense to define $s_i$ and $y_i$ in $A$ by the relations:

$Y_i=e^{\hbar y_i}$, $T_j:=s_je^{k\hbar s_j}$.

Where I'm evaluating in the representation $V$ so that the first equation makes sense, but I'm not writing that in explicitly.

One now would like to check what relations are imposed on the generators $X_i$, $y_i$ and $s_j$ so defined. Let us just check what relations we get by considering the relations of $A$ up to first non-trivial order in $\hbar$. We find:

$(T_i-q)(T_i+q^{-1})=0 \Rightarrow s_i^2=1$.

Braid relations for $T_i \Rightarrow$ braid relations for $s_i$.

$X_i$'s commute (as before). $\tilde y_i:=y_i + \sum_{i `< j} s_{ij}$.

$T_iX_iT_i=X_{i+1}\Rightarrow s_iX_is_i=X_{i+1}$

$T_iY_{i+1}T_i=Y_i\Rightarrow s_iy_is_i=y_{i+1}+s_i$

$(Y_2,X_1):=Y_2X_1Y_2^{-1}X_1^{-1}=T_1^{-2}\Rightarrow [y_2,X_1]=ks_1X_1$

which are (one form of) the relations of the trigonometric Cherednik algebra.

Further writing $X_i=e^{\hbar x_i}$, one recovers the so called rational Cherednik algebra.

So there is a hierarchy of degenerations. The top and bottom of the hierarchy are essentially symmetric in the variables $X$ and $Y$, in the precise sense that there is a "Fourier transform" automorphism swapping the variables, in both cases. Note that at the top of the hierarchy, the Fourier transform is just the order four automorphism of the elliptic curve which is the matrix $((0,1),(-1,0))$ in $PSL_2(Z)$, the mapping class group of the torus, and which can be seen in various elementary ways. At the bottom of the hierarchy, the Fourier transform is just swapping $x_i$ and $y_i$ and is related to Fourier transform of differential operators on an abelian group. In the middle, there isn't really a Fourier transform, because the symmetry was broken by degenerating the $Y_i$, but leaving the $X_i$ unscathed.


I'm not an expert on any of these things, and I can't say I've worked through the topological construction for all the cases, but I think, to answer Stephen's question, we can simply choose any lattice in $\mathbb{C}$, of rank $2$ (say $\Lambda=\mathbb{Z}1 \oplus \mathbb{Z}\mathbf{i})$, and consider $\pi_1(\mathfrak{h}_{reg}/W\ltimes\Lambda^r)$, where $r$ is the rank of $\mathfrak{h}$. Then one quotients by the relations that loops around singular points of this quotient have order 2. For instance, let me explain a nice thing that happens for BC_n type.

BC_n non-affine braid group means we configurations of 2n distinct points in $\mathbb{C}\backslash\{0}$, such that $x$ is in the configuration if and only if $-x$ is in the configuration. We don't allow zero because that we want pairs of matched points. It's not hard to see that $\pi_1$ of that configuration space is the braid group of type $BC_n$, because you can choose the repesentative of each pair lying in the upper half plane, and you get the upper half plane, except that the $r$ and $-r$ are identified for all real points, and zero is excluded. This is a punctured plane.

Now the prescription above should lead you to consider $2n$ distinct points on the elliptic curve (or rather $S^1\times S^1$; I only say elliptic curve because people sometimes mean $\mathbb{C}^\times$ by torus...). However, now you have more points to remove. Not only zero, but all half integer points would correspond to a place where $x$ and $-x$ collide. So you get the usual picture of the torus from a first course in topology, except with half-integer points removed. A very fun exercise is to work out that you can again choose the representative of each pair which lies in, say, the lower left corner of the torus, where we cut the torus in half along the diagonal from upper left to lower right. But now again you have to identify some edges, and you get .... $\mathbb{CP}^1$ with four punctures (corresponding to the four half integer points of your real torus.) Now you get to choose five parameters: one parameter for non-affine hyper planes, meaning the $T_i$, and then one for each of the new poles you've introduced. Imposing the Hecke relations on all those you get the double affine Hecke algebra of type $BC_n$ (sometimes called $C^\vee C_n$ for historical reasons) with those five parameters. Notice that this process leads you to a very different presentation of the DAHA, where you de-emphasize the lattices, and emphasize the loops around the singularities. I understand that S. Sahi gave a similar presentation for all DAHA's coming from root systems, but with probably deeper motivations than these drawings on surfaces. I'm not sure of the reference where Sahi did this.

At least in this example, one again sees the Fourier transform as just the obvious move on the real torus.

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  • $\begingroup$ If you don't mind adding a paragraph, it might be nice to say what the topological definition of the double affine braid group of a Weyl group (plus a pair of lattices) is (especially if you can make it clear why the X,Y symmetry exists in the general case)! $\endgroup$ – GS Apr 29 '10 at 14:02
  • $\begingroup$ okay i gave it a shot. I should also say that these notes: www-math.mit.edu/~etingof/73509.pdf discuss a topic of "orbifold" Hecke algebras, of which DAHA's are a special case. They also mention orbifold braid groups which are what is what you asked about. $\endgroup$ – David Jordan Apr 29 '10 at 15:04
  • $\begingroup$ Thanks! It's neat to see the parameters for the BC_n case coming in. $\endgroup$ – GS Apr 29 '10 at 15:04
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Although Vladimir already pointed a couple of relevant papers, there is also a more recent survey article by my colleague Iain Gordon on the arXiv: Symplectic reflection algebras, which you might find interesting. For one thing it has a rather extensive bibliography.

From the introduction:

Apart from confirming conjectures in algebraic combinatorics, integrable systems and real algebraic geometry, having an interesting representation theory, and connecting with noncommutative, quiver, Hall and Hopf algebras, all symplectic reflection algebras have given us is an algebraic approach to resolutions of symplectic quotient singularities.

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    $\begingroup$ I should have pointed out that this is a rather Python-esque introduction, reminiscent of the "What have the Romans ever done for us?" scene in "Life of Brian" :) $\endgroup$ – José Figueroa-O'Farrill Apr 29 '10 at 11:18
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    $\begingroup$ It's fun to read Iain's introductions and try to figure out what the inspiration is! $\endgroup$ – GS Apr 29 '10 at 11:21
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  1. For how you can explain the precise relationship between Cherednik algebras and stuff from classical algebraic geometry/commutative algebra, you can, for example, check out papers http://arxiv.org/abs/math/0407516 and http://arxiv.org/abs/math/0410293.

  2. Similarly to how the Weyl relation pq-qp=1 is the only relation between $x$ and $\frac{\partial\phantom{x}}{\partial{x}}$, defining relations for (say, rational) Cherednik algebras are relations satisfied by the symmetric group, operators of multiplication by $x_i$, and Dunkl differential-difference operators $D_i$. So one of the reason to study Cherednik algebras is the Dunkl operators are interesting. Why are they interesting? Because they commute, and can be used to prove complete integrability of some generalization of the Calogero-Moser system.

There is much more to say than a couple of quick lines I come up with, but I hope that gives some flavour...

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