Gelfand-Tsetlin modules over $\mathfrak{gl}(n)$ with arbitrary characters

et $n,m\in\mathbb{N}$. We write $\llbracket n,m\rrbracket=\{k\in\mathbb{N}\mid n\leq k\leq m\}$ and $\llbracket n\rrbracket=\llbracket 1,n\rrbracket$. We denote by $S_{n}$ the symmetric group on $n$ elements. Recall that for each $\sigma\in S_{n}$ the length of $\sigma$, denoted by $\ell(\sigma)$, is the number of inversions of $\sigma$, i.e. the number of pairs $(i,j)\in\llbracket n\rrbracket^{2}$ such that $i<j$ but $\sigma(i)>\sigma(j)$. There exists a unique longest word $w_{0}\in S_{n}$ such that $\ell(w_{0})=\frac{n(n-1)}{2}$. Also $\ell(\sigma)=\ell(\sigma^{-1})$, and $\ell(\sigma^{-1}w_{0})=\ell(w_{0})-\ell(\sigma)$. For each $i\in\llbracket n-1\rrbracket$ the $i$-th simple transposition is $s_{i}=(i,i+1)\in S_{n}$. Simple transpositions generate $S_{n}$, and the length of $\sigma\in S_{n}$ is the minimal $l$ such that $\sigma$ can be written as $s_{i_{1}}s_{i_{2}}\cdots s_{i_{l}}$. Any such writing is called a reduced decomposition of $\sigma$.

Compositions. Recall that a composition of $n$ is a sequence $\mu=(\mu_{1},\ldots,\mu_{r})$ of positive integers such that $\sum_{i=1}^{r}\mu_{i}=n$. The $\mu_{k}$ are called the parts of $\mu$. Now let $\mu=(\mu_{1},\ldots,\mu_{r})$ be a composition of $n$. For each $k\in\llbracket r\rrbracket$ set $\alpha_{k}=\alpha_{k}(\mu)=\sum_{j=1}^{k-1}\mu_{j}+1$ and $\beta_{k}=\beta_{k}(\mu)=\sum_{j=1}^{k}\mu_{j}$, so the interval $\llbracket\alpha_{k},\beta_{k}\rrbracket$ has $\mu_{k}$ elements; we refer to this interval as the $k$-th block of $\mu$.

Denote by $S_{\mu}\subset S_{n}$ the subgroup of bijections $\sigma$ such that $\sigma(\llbracket\alpha_{k},\beta_{k}\rrbracket)=\llbracket\alpha_{k},\beta_{k}\rrbracket$ for each $k\in\llbracket r\rrbracket$. This is a parabolic subgroup of $S_{n}$ in the sense of [BB-coxeter-book]*section 2.4, and we review some of the properties discussed there. By definition each $\sigma\in S_{\mu}$ is a product of the form $\sigma=\sigma^{(1)}\sigma^{(2)}\cdots\sigma^{(r)}$ with each $\sigma^{(j)}$ the identity on each $\mu$-block except the $j$-th. The length of an element on $S_{\mu}$ is $\ell(\sigma)=\ell(\sigma^{(1)})+\cdots+\ell(\sigma^{(r)})$, and $S_{\mu}$ has a unique longest word $w_{\mu}$ with $w_{\mu}^{(k)}$ the longest word of the permutation group of $\llbracket\alpha_{k},\beta_{k}\rrbracket$. We set $\mu!=\#S_{\mu}=\mu_{1}!\mu_{2}!\cdots\mu_{r}!$.

Put $\Sigma(\mu)=\{(k,i)\mid k\in\llbracket r\rrbracket,i\in\llbracket\mu_{k}\rrbracket\}$ and let $\gamma_{\mu}:\Sigma(\mu)\longrightarrow\llbracket n\rrbracket$ given by $\gamma_{\mu}(k,i)=i+\sum_{j=1}^{k-1}\mu_{j}$. This map is a bijection, and through it $S_{\mu}$ acts on $\Sigma(\mu)$. We will often identify a permutation $\sigma\in S_{\mu}$ by its action on $\Sigma(\mu)$; for example, we denote by $s_{i}^{(k)}$ the simple transposition in $S_{\mu}$ which acts on $\Sigma(\mu)$ by interchanging $(k,i)$ and $(k,i+1)$, leaving all other elements fixed. With this notation, $\sigma^{(k)}$ leaves all elements of the form $(j,l)$ with $j\neq k$ fixed.

Set

$\displaystyle\operatorname{sym}_{\mu}$

$\displaystyle=\frac{1}{\mu!}\sum_{\sigma\in S_{\mu}}\sigma;$

$\displaystyle\operatorname{asym}_{\mu}$

$\displaystyle=\frac{1}{\mu!}\sum_{\sigma\in S_{\mu}}\operatorname{sg}(\sigma)\sigma.$

These are idempotent elements of the group algebra $\mathbb{C}[S_{\mu}]$ and given a $\mathbb{C}[S_{\mu}]$-module $V$, multiplication by $\operatorname{sym}_{\mu}$, resp. $\operatorname{asym}_{\mu}$, is the projection onto the symmetric, resp. antisymmetric, component of $V$.

Refinements. A refinement of $\mu$ is a collection of compositions $\eta=(\eta^{(1)},\ldots,\eta^{(r)})$ with each $\eta^{(k)}$ a composition of $\mu_{k}$. If $\eta$ is a refinement of $\mu$ then the concatenation of the $\eta^{(k)}$’s is also a composition of $n$, which by abuse of notation we will also denote by $\eta$.

If $\eta$ refines $\mu$ then $S_{\eta}\subset S_{\mu}$. We say that $\sigma\in S_{\mu}$ is a $\eta$-shuffle if it is increasing in each $\eta$-block. Among the elements of a coclass $\sigma S_{\mu}\in S_{\mu}/S_{\eta}$ there is exactly one $\eta$-shuffle, and this is the unique element of minimal length in the coclass. We denote the set of all $\eta$-shuffles in $S_{\mu}$ by $\mathsf{Shuffle}^{\mu}_{\eta}$. The group $S_{\eta}$ acts on $\Sigma(\mu)$ by restriction, and the orbits of this action are also called the $\eta$-blocks of $\mu$.

mu-points We write $\mathbb{C}^{\mu}=\mathbb{C}^{\mu_{1}}\oplus\mathbb{C}^{\mu_{2}}\oplus\cdots\oplus\mathbb{C}^{\mu_{r}}$; thus $v\in\mathbb{C}^{\mu}$ is an $r$-uple of vectors $(v_{1},\ldots,v_{r})$ with $v_{k}\in\mathbb{C}^{\mu_{k}}$. We refer to the elements of $\mathbb{C}^{\mu}$ as $\mu$-points, or simply points if the composition $\mu$ is fixed. For each $(k,i)\in\Sigma(\mu)$ we write $v_{k,i}$ for the $i$-th coordinate of $v_{k}$. We refer to the $v_{k}$’s as the $\mu$-blocks of $v$, and to the $v_{k,i}$’s as the entries of $v$. We say that a $\mu$-point $v$ is integral if all its entries lie in $\mathbb{Z}$. Clearly $\mathbb{C}^{\mu}$ is an affine variety, and we denote by $\mathbb{C}[X_{\mu}]$ the polynomial algebra generated by $x_{k,i}$ with $(k,i)\in\Sigma(\mu)$, which is the coordinate ring of $\mathbb{C}^{\mu}$. We also denote by $\mathbb{C}(X_{\mu})$ the field of fractions of $\mathbb{C}[X_{\mu}]$. The group $S_{\mu}$ acts on $\mathbb{C}^{\mu}$ in an obvious way, and this induces actions on $\mathbb{C}[X_{\mu}]$ and $\mathbb{C}(X_{\mu})$.

If $\eta$ is a refinement of $\mu$ then there exists an isomorphism $\mathbb{C}^{\eta}\cong\mathbb{C}^{\mu}$, so we can talk of the $\eta$-blocks of $v\in\mathbb{C}^{\mu}$. Since $\eta$ refines $\mu$ we get an inclusion $S_{\eta}\subset S_{\mu}$, and so $S_{\eta}$ acts on $\mathbb{C}^{\mu}$ by restriction. The isomorphism $\mathbb{C}^{\eta}\cong\mathbb{C}^{\mu}$ is $S_{\eta}$-equivariant.

D:gt-module For each $k\in\llbracket n\rrbracket$ we denote by $U_{k}$ the enveloping algebra of $\mathfrak{gl}(k,\mathbb{C})$, and set $U=U_{n}$. Inclusion of matrices in the top left corner induces a chain

$\displaystyle\mathfrak{gl}(1,\mathbb{C})\subset\mathfrak{gl}(2,\mathbb{C})\subset\cdots\subset\mathfrak{gl}(n,\mathbb{C}),$

which in turn induces a chain $U_{1}\subset U_{2}\subset\cdots\subset U_{n}$. Denote by $Z_{k}$ the center of $U_{k}$ and by $\Gamma$ the subalgebra of $U$ generated by $\bigcup_{k=1}^{n}Z_{k}$. This algebra is the Gelfand-Tsetlin subalgebra of $U$, and it is generated by the elements

$\displaystyle c_{k,i}$

$\displaystyle=\sum_{(r_{1},\ldots,r_{i})\in\llbracket k\rrbracket^{i}}E_{r_{1},r_{2}}E_{r_{2},r_{3}}\cdots E_{r_{i},r_{1}},$

$\displaystyle(k,i)\in\Sigma(\mu).$

By work of Zhelobenko there exists an isomorphism $\iota:\Gamma\longrightarrow\mathbb{C}[X_{\mu}]^{S_{\mu}}$, given by $\iota(c_{k,i})=\gamma_{k,i}$ where

$\displaystyle\gamma_{k,i}$

$\displaystyle=\sum_{j=1}^{k}(x_{k,j}+k-1)^{i}\prod_{m\neq j}\left(1-\frac{1}{x_{k,j}-x_{k,m}}\right),$

see [FGR-1-singular]*subsection 3.1 for details. It follows that $\operatorname{Specm}\Gamma\cong\mathbb{C}^{\mu}/S_{\mu}$, and so every $\mu$-point $v$ induces a character $\chi_{v}:\Gamma\longrightarrow\mathbb{C}$ by setting $c\in\Gamma\mapsto\iota(c)(v)$, and two $\mu$-points induce the same character if and only if they lie in the same $S_{\mu}$-orbit.

Let $M$ be a Gelfand-Tsetlin module. Identifying $\mathfrak{m}$ with the character $\chi:\Gamma\longrightarrow\Gamma/\mathfrak{m}\cong\mathbb{C}$, we also write $M[\chi]$ for $M[\mathfrak{m}]$. We say that $\chi$ is a Gelfand-Tsetlin character of $M$ if $M[\chi]\neq 0$ and define the multiplicity of $\chi$ in $M$ as $\dim_{\mathbb{C}}M[\chi]$. The Gelfand-Tsetlin support of $M$ is the set of all of its Gelfand-Tsetlin characters. We will often abreviate Gelfand-Tsetlin for GT, or ommit it completely when it is clear from the context.

T:gt-tableaux Gelfand-Tsetlin tableaux. For the rest of this document we denote by $\mu$ the composition $(1,2,\ldots,n)$ of $N$. To each $\mu$-point we assign a triangular array

Such an array is known as a Gelfand-Tsetlin tableau. With this identification the group $S_{\mu}$ acts on the space of all tableaux. The main difference between the space of $\mu$-points and the space of all tableaux is that the first is obviously a $\mathbb{C}$-vector-space, while the second one is not. In particular $T(v+w)\neq T(v)+T(w)$, since the second expression does not make sense (though we will eventually consider a vector-space generated by tableaux).

A standard $\mu$-point is one in which $v_{k,i}-v_{k-1,i}\in\mathbb{Z}_{\geq 0}$ and $v_{k-1,i}-v_{k,i+1}\in\mathbb{Z}_{>0}$ for all $1\leq i<k\leq n$. We denote by $\mathbb{C}^{\mu}_{\mathsf{std}}$ the set of all standard $\mu$-points. Notice that given $\lambda=(\lambda_{1},\ldots,\lambda_{n})$, there exists finitely many standard tableaux with top row equal to $\lambda-(0,1,2,\ldots,n-1)$, and that this number is nonzero if and only if $\lambda_{i}-\lambda_{i+1}\in\mathbb{Z}_{\geq 0}$, i.e. $\lambda$ must be a dominant integral weight of $\mathfrak{gl}(n,\mathbb{C})$. This definition was introduced by Gelfand and Tsetlin in their article [GT-modules] in order to give an explicit presentation of irreducible $\mathfrak{gl}(n,\mathbb{C})$-modules.

The $U$-module $V(\lambda)$ is an irreducible finite dimensional representation of maximal weight $\lambda$, so this theorem provides an explicit presentation of all finite dimensional simple $\mathfrak{gl}(n,\mathbb{C})$-modules. The last statement of the theorem, giving the action of the generators of $\Gamma$ on $V(\lambda)$, is due to Zhelobenko.

iven $1\leq i\leq k<n$ we set

$\displaystyle e_{k,i}^{+}$

$\displaystyle=\frac{\prod_{j=1}^{k+1}(x_{k,i}-x_{k+1,j})}{\prod_{j\neq i}(x_{k,i}-x_{k,j})},$

$\displaystyle e_{k,i}^{-}$

$\displaystyle=\frac{\prod_{j=1}^{k-1}(x_{k,i}-x_{k-1,j})}{\prod_{j\neq i}(x_{k,i}-x_{k,j})},$

which are elements of $\mathbb{C}(X_{\mu})$. These are the rational functions that appear in Gelfand and Tsetlin’s presentation of finite dimensional irreducible $U$-modules, and we refer to them as the Gelfand-Tsetlin functions.

A $\mu$-point $v$, or equivalently the corresponding GT tableaux, is called generic if $v_{k,i}-v_{k,j}\notin\mathbb{Z}$ for all $1\leq i<j\leq k<n$, otherwise it is called singular. Notice that a generic tableau may have entries in the top row whose difference is an integer.

Let $\mathbb{Z}^{\mu}_{0}$ be the set of all integral $\mu$-points with $z_{n,i}=0$ for all $1\leq i\leq n$ (i.e., the corresponding GT tableau has its top row filled with zeros). If $v$ is generic then the set $\{v+z\mid z\in\mathbb{Z}^{\mu}_{0}\}$ contains no poles of the Gelfand-Tsetlin functions. This idea was used by Drozd, Futorny and Ovsienko in [DFO-GT-modules] to give a $U$-module structure to the vector-space

$\displaystyle V(T(v))$

$\displaystyle=\langle T(v+z)\mid z\in\mathbb{Z}^{\mu}_{0}\rangle_{\mathbb{C}}.$

Since the action of $U$ is given by the Gelfand-Tsetlin functions, each tableau $T(v)$ is an eigenvector of $\Gamma$ and hence $V(T(v))$ is a GT module with support $\{\chi_{v+z}\mid z\in\mathbb{Z}^{\mu}_{0}\}$ and all multiplicities equal to $1$.

big-module The main goal of this article is to give a nontrivial $U$-module structure to the space $V(T(v))$ for an arbitrary $\mu$-point $v$. The approach from [DFO-GT-modules] breaks down if $v$ is not generic, but Futorny, Grantcharov and the first named author extended this construction to a subset of singular characters in [FGR-1-singular, FGR-2-index], and in this article we extend their work to arbitrary $\mu$-points. In order to do this, we introduce a large $U$-module which serves as a formal model of a GT module.

Set $K=\mathbb{C}(X_{\mu})$ to be the field of rational functions over $\mu$-points. As mentioned above $e_{k,i}^{\pm}\in K$ for all $1\leq i\leq k<n$. We set $V_{\mathbb{C}}$ to be the $\mathbb{C}$-vector-space with basis $\left\{T(z)\mid z\in\mathbb{Z}^{\mu}_{0}\right\}$, and $V_{K}=K\otimes_{\mathbb{C}}V_{\mathbb{C}}$. Since $S_{\mu}$ acts naturally on both $K$ and $V_{\mathbb{C}}$, it acts on $V_{K}$ by the diagonal action, i.e. given $\sigma\in S_{\mu},f\in K$ and $z\in\mathbb{Z}^{\mu}_{0}$ the action is the linear extension of $\sigma\cdot f\otimes T(z)=\sigma(f)\otimes T(\sigma(z))$. The group $\mathbb{Z}^{\mu}_{0}$ also acts on $\mathbb{C}[X_{\mu}]$, with $\delta^{k,i}\cdot x_{l,j}=x_{l,j}+\delta_{k,l}\delta_{i,j}$. This action extends to $K$, and given $f\in K,z\in\mathbb{Z}^{\mu}_{0}$ we sometimes write $f(x+z)$ instead of $z\cdot f$. The actions of $S_{\mu}$ and $\mathbb{Z}^{\mu}_{0}$ do not commute, since $\sigma(z\cdot f)=\sigma(z)\cdot\sigma(f)$.

• Proof.

  The proof that $V_{K}$ is a $U$-module is identical to the proof of [Zad-1-sing]*Proposition 1. It follows from the definitions that $\sigma\cdot e^{\pm}_{k,i}=e^{\pm}_{k,\sigma^{(k)}(i)}$, and using this it is easy to check that the canonical generators of $U$ act by $S_{\mu}$-equivariant operators. ∎

We will refer to the $U$-module $V_{K}$ as the “big GT module”. The big GT modulewas introduced in [Zad-1-sing], where it was proved that certain sublattices can serve as universal models for modules with generic or $1$-singular characters ($1$-singular here means that there is at most one pair of entries in the same $\mu$-block differing by an integer). In the following sections of this article we will extend this argument to arbitrary characters without any restriction.

sdd-intro Throughout this section we fix $m\in\mathbb{N}$ and $\eta=(\eta_{1},\ldots,\eta_{r})$ a composition of $m$. Recall that $\mathbb{C}[X_{\eta}]=\mathbb{C}[x_{k,i}\mid(k,i)\in\Sigma(\eta)]$, on which $S_{\eta}$ acts by $\sigma(x_{k,i})=x_{\sigma\cdot(k,i)}$. We set $F=\mathbb{C}(X_{\eta})$, the fraction field of $\mathbb{C}[X_{\eta}]$, with its obvious $S_{\eta}$-action.

Set

$\displaystyle\Delta_{k}$

$\displaystyle=\prod_{1\leq i<j\leq\eta_{k}}(x_{k,i}-x_{k,j}),$

$\displaystyle\Delta_{\eta}$

$\displaystyle=\prod_{k=1}^{r}\Delta_{k}.$

If $f\in\mathbb{C}[X_{\eta}]$ is a polynomial such that $\sigma(f)=\operatorname{sg}(\sigma)f$ for all $\sigma\in S_{\eta}$ then $f=g\Delta_{\eta}$, with $g$ an $S_{\eta}$-invariant polynomial.

ddoperators Divided difference operators. Since the action of $S_{\eta}$ extends to $F$ we can form the smash product $F\#S_{\eta}$, which is the $\mathbb{C}$-algebra whose underlying vector-space is $F\otimes\mathbb{C}[S_{\eta}]$ and product given by $(f\otimes\sigma)\cdot(g\otimes\tau)=f\sigma(g)\otimes\sigma\tau$ for all $f,g\in F$ and all $\sigma,\tau\in S_{\eta}$. We will usually ommit the tensor product symbol when writing elements in $F\#S_{\eta}$, so $f\sigma$ stands for $f\otimes\sigma$. We must be careful to distinguish the action of $\sigma\in S_{\eta}$ on a rational function $f\in F$, denoted by $\sigma(f)$, from their product in $F\#S_{\eta}$, which is $\sigma\cdot f=\sigma(f)\sigma$.

Recall that for each $(k,i)\in\Sigma(\eta)$ we denote by $s_{i}^{(k)}$ the unique simple transposition in $S_{\eta}$ which acts on $\Sigma(\eta)$ by interchaging $(k,i)$ and $(k,i+1)$, while leaving the other elements fixed. The divided difference associated to $s_{i}^{(k)}$ is $\partial_{i}^{(k)}=\frac{1}{x_{k,i}-x_{k,i+1}}(\mathsf{id}-s_{i}^{(k)})\in F\#S_{\eta}$. These elements satisfy the relations

	$\displaystyle(\partial_{i}^{(k)})^{2}$	$\displaystyle=0$
	$\displaystyle\partial_{i}^{(k)}\partial_{j}^{(l)}$	$\displaystyle=\partial_{j}^{(l)}\partial_{i}^{(k)}$	$\displaystyle\mbox{ if }l\neq k\mbox{ or }|i-j|>1;$
	$\displaystyle\partial_{i}^{(k)}\partial_{i+1}^{(k)}\partial_{i}^{(k)}$	$\displaystyle=\partial_{i+1}^{(k)}\partial_{i}^{(k)}\partial_{i+1}^{(k)}$	$\displaystyle\mbox{ for }1\leq i\leq k-2.$

Let $\sigma=\sigma^{(1)}\cdots\sigma^{(n)}\in S_{\eta}$. For each $k\in\llbracket n\rrbracket$ we can write $\sigma^{(k)}$ as a reduced composition $s_{i_{1}}^{(k)}s_{i_{2}}^{(k)}\cdots s_{i_{l}}^{(k)}$, with $\ell(\sigma^{(k)})=l$. We set $\partial_{\sigma}^{(k)}=\partial_{i_{1}}^{(k)}\cdot\partial_{i_{2}}^{(k)}\cdots\partial_{i_{l}}^{(k)}$, and $\partial_{\sigma}=\partial_{\sigma}^{(1)}\cdot\partial_{\sigma}^{(2)}\cdots\partial_{\sigma}^{(n)}$; the relations among the divided difference operators imply that $\partial_{\sigma}^{(k)}$, and hence $\partial_{\sigma}$, is independent of the reduced composition we choose for $\sigma^{(k)}$.

The equality $\ell(\sigma)+\ell(\tau)=\ell(\sigma\tau)$ holds if and only if the concatenation of a reduced composition of $\sigma$ with a reduced composition of $\tau$ is a reduced composition of $\sigma\tau$, and this implies that $\partial_{\sigma}\cdot\partial_{\tau}=\partial_{\sigma\tau}$. If equality does not hold then the concatenation is not reduced, which implies that in the product $\partial_{\sigma}\cdot\partial_{\tau}$ there must be a term of the form $\partial_{i}^{(k)}\cdot\partial_{i}^{(k)}=0$. Thus for all $\sigma,\tau\in S_{\eta}$ we get.

$\displaystyle\partial_{\sigma}\cdot\partial_{\tau}$

$\displaystyle=\begin{cases}\partial_{\sigma\tau}&\mbox{ if }\ell(\sigma)+\ell(\tau)=\ell(\sigma\tau);\\ 0&\mbox{otherwise.}\end{cases}$

L:dd-algebra Divided differences are usually defined as operators on the polynomial algebra $\mathbb{C}[X_{\eta}]$, although they make sense over any $F$-vector-space with an equivariant $S_{\eta}$-action. Since these operators play an important role in the sequel, we gather some of their basic properties in the following lemma.

• Proof.

  Item 1 is an easy computation following from the definition. Now set $s=s_{i}^{(k)}$ and $\partial_{s}=\partial_{i}^{(k)}$. By definition $\partial_{s}\cdot s=-\partial_{s}$ and $s\cdot\partial_{s}=\partial_{s}$, and since $w_{\eta}=(w_{\eta}s)s$ with $\ell(w_{\eta}s)=\ell(w_{\eta})-1$ we get

  $\displaystyle\partial_{w_{\eta}}\cdot s$

  $\displaystyle=\partial_{w_{\eta}s}\cdot\partial_{s}\cdot s=-\partial_{w_{\eta}s}\cdot\partial_{s}=-\partial_{w_{\eta}}.$

  This along with the previous item gives

  	$\displaystyle 0$	$\displaystyle=\partial_{w_{\eta}}\cdot(\partial_{s}\cdot f)=\partial_{w_{\eta}}\cdot s(f)\partial_{s}+\partial_{w_{\eta}}\cdot\partial_{s}(f)$
  		$\displaystyle=\partial_{w_{\eta}}\cdot(s\cdot fs)\cdot\partial_{s}+\partial_{w_{\eta}}\cdot\partial_{s}(f)=-\partial_{w_{\eta}}\cdot f\partial_{s}+\partial_{w_{\eta}}\cdot\partial_{s}(f)$

  which proves that item 2 holds for $\sigma=s$. Since $k$ and $i$ are arbitrary, the general case follows by induction on the length of $\sigma$.

  A similar argument as above shows that $\sigma\cdot\partial_{w_{\eta}}=\partial_{w_{\eta}}$ for all $\sigma\in S_{\eta}$. Induction on the length of $\sigma$ shows that $\partial_{\sigma}=\sum_{\tau\in S_{\eta}}\frac{1}{f_{\tau,\sigma}}\tau$ with $f_{\tau,\sigma}\in\mathbb{C}[X_{\eta}]$ homogeneous of degree $\ell(\sigma)$. If we put $f_{\tau}=f_{\tau,w_{\eta}}$, the equalities $\sigma\cdot\partial_{w_{\eta}}=\partial_{w_{\eta}}$ and $\partial_{w_{\eta}}\cdot\sigma=\operatorname{sg}(\sigma)\partial_{w_{\eta}}$ imply that $f_{\sigma}=\sigma(f_{e})=\operatorname{sg}(\sigma)f_{e}$, so $f_{e}$ is an $S_{\eta}$-antisymmetric polynomial of degree $\ell(w_{\eta})$, i.e. a scalar multiple of $\Delta_{\eta}$. Now $\partial_{w_{\eta}}(\Delta_{\eta})=\eta!$, so $f_{e}=\frac{\eta!}{\Delta_{\eta}}$ and

  $\displaystyle\partial_{w_{\eta}}$

  $\displaystyle=\frac{\eta!}{\Delta_{\eta}}\sum_{\sigma\in S_{\eta}}\operatorname{sg}(\sigma)\sigma=\eta!\sum_{\sigma\in S_{\eta}}\sigma\cdot\frac{1}{\Delta_{\eta}}.$

  This completes the proof of item 3. ∎

Many subalgebras of $F$ are stable by the action of divided differences. It is a well-known fact that this is the case for $\mathbb{C}[X_{\eta}]$. Assume $\mathbb{C}[X_{\eta}]\subset A\subset F$ is closed under the action of $S_{\eta}$. Then any rational function in $A$ can be written as a quotient $p/q$ with $p,q\in\mathbb{C}[X_{\eta}]$ and $q$ $S_{\eta}$-invariant, so $\partial_{\sigma}(f)=\partial_{\sigma}(p/q)=\partial_{\sigma}(p)/q\in A$ for each $\sigma\in S_{\eta}$ and $A$ is also closed under divided differences.

polynomial-dd We focus now on the action of divided differences over the polynomial ring $\mathbb{C}[X_{\eta}]$. First, for every $\sigma\in S_{\eta}$ the operator $\partial_{\sigma}:\mathbb{C}[X_{\eta}]\longrightarrow\mathbb{C}[X_{\eta}]$ is homogeneous of degree $-\ell(\sigma)$. Now let $\mathfrak{p}_{\eta}$ be the ideal generated by $\{x_{k,i}-x_{k,j}\mid(k,i),(k,j)\in\Sigma(\eta)\}$, and let $\mathbb{C}[\mathfrak{p}_{\eta}]\subset\mathbb{C}[X_{\eta}]$ be the algebra generated by $\mathfrak{p}_{\eta}$; clearly $\Delta_{\eta}\in\mathfrak{p}_{\eta}$. Since $\partial_{i}^{(k)}(x_{l,j}-x_{l,t})\in\mathbb{Z}$, the algebra $\mathbb{C}[\mathfrak{p}_{\eta}]$ is stable by the action of divided differences. It follows that $\partial_{\sigma}\Delta_{\eta}\in\mathfrak{p}_{\eta}$ if $\ell(\sigma)<\deg\Delta_{\eta}=\eta!$, while $\partial_{w_{\eta}}\Delta_{\eta}=\eta!$.

Let $\sigma,\tau\in S_{\eta}$. By item 2 of Lemma LABEL:L:dd-algebra we have

$\displaystyle\frac{1}{\eta!}\partial_{w_{\eta}}((\partial_{\tau}\Delta_{\eta})(\partial_{\sigma}\Delta_{\eta}))$

$\displaystyle=\frac{1}{\eta!}\partial_{w_{\eta}}(\Delta(\partial_{\tau^{-1}}\partial_{\sigma}\Delta_{\eta}))=\operatorname{sym}_{\eta}(\partial_{\tau^{-1}}\partial_{\sigma}\Delta_{\eta}).$

Now if $\ell(\tau^{-1})+\ell(\sigma)\geq\ell(w_{\eta})$ this is $0$ unless $\tau^{-1}\sigma=w_{\eta}$, in which case it equals $\eta!$. If $\ell(\tau^{-1})+\ell(\sigma)<\ell(w_{\eta})$ then the best we can say is that this element lies in $\mathfrak{p}_{\eta}^{S_{\eta}}$.