What are the generators of the degenerate affine Hecke algebra $H(k)$ for $k > 0$?
Degenerate Affine Hecke Algebra – Key Concepts and Applications
hecke-algebrasrt.representation-theory
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Well the first thing to say is to look at the very enthusiastic and world-encompassing papers of Cherednik himself on DAHA as the center of the mathematical world (say his 1998 ICM). I'll mention a couple of more geometric aspects, but this is a huuuge area..
There are at least three distinct geometric appearances of DAHA, which you could classify by the number of loops (as in loop groups) that appear - two, one or zero. (BTW for those in the know I will mostly intentionally ignore the difference between DAHA and its spherical subalgebra.)
Double loop picture: See e.g. Kapranov's paper arXiv:math/9812021 (notes for lectures of his on it available on my webpage) and the related arXiv:math/0012155. The intuitive idea, very hard to make precise, is that DAHA is the double loop (or 2d local field, such as F_q((s,t)) ) analog of the finite (F_q) and affine (F_q((s)) ) Hecke algebras. In other words it appears as functions on double cosets for the double loop group and its "Borel" subalgebra. (Of course you need to decide what "functions" or rather "measures" means and what "Borel" means..) This means in particular it controls principal series type reps of double loop groups, or the geometry of moduli of G-bundles on a surface, looked at near a "flag" (meaning a point inside a curve inside the surface). The rep theory over 2d local fields that you would need to have for this to make sense is studied in a series of papers of Kazhdan with Gaitsgory (arXiv:math/0302174, 0406282, 0409543), with Braverman (0510538) and most recently with Hrushovski (0510133 and 0609115). The latter is totally awesome IMHO, using ideas from logic to define definitively what measure theory on such local fields means.
Single loop picture: Affine Hecke algebras have two presentations, the "standard" one (having to do with abstract Kac-Moody groups) and the Bernstein one (having to do specifically with loop groups). These two appear on the two sides of Langlands duality (cf eg the intro to the book of Chriss and Ginzburg). Likewise there's a picture of DAHA that's dual to the above "standard" one. This is developed first in Garland-Grojnowski (arXiv:q-alg/9508019) and more thoroughly by Vasserot arXiv:math/0207127 and several papers of Varagnolo-Vasserot. The idea here is that DAHA appears as the K-group of coherent sheaves on G(O)\G(K)/G(O) - the loop group version of the Bruhat cells in the finite flag manifold (again ignoring Borels vs parabolics). Again this is hard to make very precise. This gives in particular a geometric picture for the reps of DAHA, analogous to that for AHA due to Kazhdan-Lusztig (see again Chriss-Ginzburg).
[EDIT: A new survey on this topic by Varagnolo-Vasserot has just appeared.]
Here is where geometric Langlands comes in: the above interp means that DAHA is the Hecke algebra that acts on (K-groups of) coherent sheaves on T^* Bun_G X for any Riemann surface X -- it's the coherent analog of the usual Hecke operators in geometric Langlands. Thus if you categorify DAHA (look at CATEGORIES of coherent sheaves) you get the Hecke functors for the so-called "classical limit of Langlands" (cotangent to Bun_G is the classical limit of diffops on Bun_G).
The Cherednik Fourier transform gives an identification between DAHA for G and the dual group G'. In this picture it is an isom between K-groups of coherent sheaves on Grassmannians for Langlands dual groups (the categorified version of this is conjectured in Bezrukavnikov-Finkelberg-Mirkovic arXiv:math/0306413). This is a natural part of the classical limit of Langlands: you're supposed to have an equivalence between coherent sheaves on cotangents of Langlands dual Bun_G's, and this is its local form, identifying the Hecke operators on the two sides!
In this picture DAHA appears recently in physics (since geometric Langlands in all its variants does), in the work of Kapustin (arXiv:hep-th/0612119 and with Saulina 0710.2097) as "Wilson-'t Hooft operators" --- the idea is that in SUSY gauge theory there's a full DAHA of operators (with the above names). Passing to the TFT which gives Langlands kills half of them - a different half on the two sides of Langlands duality, hence the asymmetry.. but in the classical version all the operators survive, and the SL2Z of electric-magnetic/Montonen-Olive S-duality is exactly the Cherednik SL2Z you mention..
Finally (since this is getting awfully long), the no-loop picture: this is the one you referred to in 2. via Dunkl type operators. Namely DAHA appears as difference operators on H/W (and its various degenerations, the Cherednik algebras, appear by replacing H by h and difference by differential). In this guise (and I'm not giving a million refs to papers of Etingof and many others since you know them better) DAHA is the symmetries of quantum many-body systems (Calogero-Moser and Ruijsenaars-Schneiders systems to be exact), and this is where Macdonald polynomials naturally appear as the quantum integrals of motion. The only thing I'll say here is point to some awesome recent work of Schiffmann and Vasserot arXiv:0905.2555, where this picture too is tied to geometric Langlands.. very very roughly the idea is that H/W is itself (a degenerate version of an open piece of) a moduli of G-bundles, in the case of an elliptic curve. Thus studying DAHA is essentially studying D-modules or difference modules on Bun_G in genus one (see Nevins' paper arXiv:0804.4170 where such ideas are developed further). Schiffman-Vasserot show how to interpret Macdonald polynomials in terms of geometric Eisenstein series in genus one.. enough for now.
It would be great if they existed, even if only for the symmetric group. One possible application would be to algebraic combinatorics: Mark Haiman has collected data suggesting that the ring $R/R^{S_n}_+$ has dimension $2^n (n+1)^{n-2}$, where $$R=\mathbb{Q}[x_1,\dots,x_n,y_1,\dots,y_n,z_1,\dots,z_n],$$ the symmetric group acts by simultaneously permuting the three sets of variables, and the notation $R^{S_n}_+$ means the ideal generated by positive degree symmetric polynomials.
Iain Gordon showed that it is possible to use the representation theory of the rational Cherednik algebra (the rational object in the rational/trigonometric/elliptic trichotomy whose elliptic object is the DAHA) to establish the correct lower bound on this dimension in the case of two sets of variables (this dimension turns out to be $(n+1)^{n-1}$, a theorem proved by Haiman using a suggestion of Procesi and the geometry of the Hilbert scheme of points in the plane). So one might hope to use the representation theory of a TAHA to prove the correct lower bound, at least.
The reason the double affine Hecke algebra exists at all is a little subtle, and has to do with @Theo Johnson-Freyd's comments to the question: the affine Hecke algebra has two realizations. First, the affine Hecke algebra is an affinization of a finite Hecke algebra; second, it is the Hecke algebra associated to the affine Weyl group (or, if you prefer, for a certain specialization it is the Hecke algebra corresponding to an Iwahori subgroup of a p-adic group). Starting with the second presentation, one affinizes again to obtain the DAHA. The point here is that
$$\{\text{affine Hecke algebras} \}=\{\text{Hecke algebras of affine groups} \}$$
and we know how to affinize the Hecke algebras on the RHS.
But so far the DAHA has no second realization as the Hecke algebra of something that can be affinized again. Perhaps recent work of Kazhdan and his collaborators could help here, but I have not read these papers carefully enough to know.
There is a second approach that is somewhat more geometrical. The rational Cherednik algebra is a deformation of the algebra $\mathbb{Q}[x_1,\dots,x_n,y_1,\dots,y_n] \rtimes S_n$, so one might look for nice deformations of the analogous objects in three sets of variables. Perhaps experts in Hochschild cohomology have done calculations suggesting where to look?
Best Answer
The degenerate affine Hecke algebra $H(k)$ over a field $F$ is isomorphic (as a vector space) to the tensor product $$ H(k)=^{\mathrm{v.s.}} FS_k \otimes F[x_1,\ldots,x_k] $$ where $FS_k$ is the group algebra of the symmetric group generated by simple reflections $s_1,\ldots,s_{k-1}$ ($s_i=(i,i+1)$) and $F[x_1,\ldots,x_k]$ is a polynomial algebra. Multiplication is defined so that $FS_k\otimes 1$ and $1\otimes F[x_1,\ldots,x_k]$ are subalgebras, and it is convenient to abuse notation by identifying these subalgebras with $FS_k$ and $F[x_1,\ldots,x_k]$, respectively. Finally, the mixed relations are given by $$ x_{i+1}s_i=s_ix_i +1,\;\;1\leq i\leq k-1 $$ and $$ x_js_i=s_ix_j,\;\;j\neq i,i+1. $$ In addition to being a subalgebra, $FS_k$ is a quotient of $\pi:H(k)\twoheadrightarrow FS_k$ obtained by setting $x_1=0$. Note that the first mixed relation can be rewritten as $$ x_{i+1}=s_ix_is_i + s_i $$ so we can write $$x_i=s_{i-1}\cdots s_1 x_1 s_1\cdots s_{i-1} +L_i$$ where $L_i$ is the $i$th Jucys-Murphy element. In particular, $\pi(x_i)=L_i$.
I recommend Kleshchev's book `Linear and Projective Representations of Symmetric groups' for further information.