Question

What are the generators of the degenerate affine Hecke algebra $H(k)$ for $k > 0$?

Answer 1

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.

Answer 2

I probably misunderstand the question, but let me give a couple of comments any way.

Generators and relations of degenerate affine Hecke algebra can be found e.g. page 20 section 6 of http://arxiv.org/abs/1008.3739 A categorical approach to classical and quantum Schur-Weyl duality Alexei Davydov, Alexander Molev

They have been introduced by Drinfeld to describe Schur-Weyl duality for Yangians. (And later by Lusztig for another reasons). There is a whole bunch of SW-dualities generalizing classical one to the case of various quantum groups:

Hecke algebra <-> U_q(gl) (Jimbo)

Affine Hecke algebra <-> Affine U_q(gl)

Yangians <-> Degenerate Affine Hecke algebra

toroidal quantum group of type SL(n+1) <-> Cherednik's double affine Hecke algebra (M. Varagnolo, E. Vasserot http://arxiv.org/abs/q-alg/9506026 )

At least the first 3 are compatible in the sense that degenerating the object on the left hand side and on the right hand side we can preserve the duality.

In general I think is amazing and still not clearly understood (if it is possible at all) that Hecke algebra which originally appeared by absolutely different reason, show up themselves in SW-duality for quantum groups.