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



Degenerate affine Hecke Algebra
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.


