The degenerate affine Hecke algebra

The cases $d=0,1$. For $d=0$ there are no symbols, so ${\mathcal{F}}_0=\mathbb{C}$ and $I_0=0$, giving $H_0=\mathbb{C}$. For $d=1$ the only symbol is $x_1$ and the list of relations is empty; the words in $x_1$ are $1,x_1,x_1^2,\dots$, a $\mathbb{C}$-basis of ${\mathcal{F}}_1$ by [F4], so $H_1$ is the polynomial algebra $\mathbb{C}[x_1]$ with basis $x_1^r$, $r\ge 0$.

The rational-function model. On the $\mathbb{C}$-vector space $\mathrm{\Omega }$ of [F3] let $m_f$ denote multiplication by $f\in \mathrm{\Omega }$, and for $1\le i<d$ let ${\sigma }_i$ be the operator $({\sigma }_{i}f)(y_1,\dots ,y_d)=f(y_1,\dots ,y_{i+1},y_i,\dots ,y_d)$ interchanging the $i$-th and $(i+1)$-st arguments. Put $a_i:=d_i^{-1}\in \mathrm{\Omega }$ (invertible by [F3]), ${\delta }_i:=a_i(1+{\sigma }_i)$ and $T_i:={\sigma }_i-{\delta }_i$. For a word $w=s_{i_1}\cdots s_{i_k}$ in the generators of $S_d$ set ${\sigma }_w:={\sigma }_{i_1}\text{\hspace{0.17em}⁣}\circ \cdots \circ {\sigma }_{i_k}$ and $T_w:=T_{i_1}\text{\hspace{0.17em}⁣}\cdots T_{i_k}$. The operators ${\sigma }_w$ are well defined by the Coxeter presentation [F1], because the ${\sigma }_i$ satisfy those relations and the presentation says that any two words for the same $w$ are related by them.

The skew algebra. Let $D:=\{{\sum }_{w\in S_d}{m}_{f_w}{\sigma }_w:f_w\in \mathrm{\Omega }\}\subseteq {\mathrm{End}}_{\mathbb{C}}(\mathrm{\Omega })$, a $\mathbb{C}$-subspace. For $f,g\in \mathrm{\Omega }$ and $u,v\in S_d$ the identities ${\sigma }_u{m}_g=m_{{\sigma }_u(g)}{\sigma }_u$ and ${\sigma }_u{\sigma }_v={\sigma }_{uv}$ hold (the first expresses the substitution action on functions, the second is word concatenation), so $(m_f{\sigma }_u)(m_g{\sigma }_v)=m_{f{\sigma }_u(g)}{\sigma }_{uv}$; hence $D$ is a subalgebra of ${\mathrm{End}}_{\mathbb{C}}(\mathrm{\Omega })$. The set $\{{\sigma }_w\}$ is $\mathrm{\Omega }$-linearly independent: suppose ${\sum }_w{f}_w{\sigma }_w=0$ with $f_w\in \mathrm{\Omega }$ not all zero; multiplying the resulting identity of functions by a common denominator of the $f_w$, we may assume $f_w\in \mathbb{C}[x_1,\dots ,x_d]$. Fix integers $N_1>\cdots >N_d$ whose consecutive gaps exceed the total degree of every $f_w$, and order the finitely many $w$ with $f_w\mathrm{\ne }0$ by the lexicographic order of the vectors $w\cdot N:=(N_{w^{-1}(1)},\dots ,N_{w^{-1}(d)})$; let $w$ be maximal and let $e$ be the exponent of a top monomial of $f_w$. Applying the sum to the monomial $y^N$ gives ${\sum }_{w^{\mathrm{\prime }}}{f}_{w^{\mathrm{\prime }}}(y)y^{w^{\mathrm{\prime }}\cdot N}$, because ${\sigma }_{w^{\mathrm{\prime }}}{y}^N=y^{w^{\mathrm{\prime }}\cdot N}$; the vectors $w^{\mathrm{\prime }}\cdot N$ are distinct, and at the first coordinate where $w^{\mathrm{\prime }}\cdot N$ differs from $w\cdot N$ the gap exceeds the degree bound, so the monomial $y^{w\cdot N+e}$ occurs exactly once, in the summand $f_w{y}^{w\cdot N}$, with nonzero coefficient. This contradicts the vanishing of the sum, proving the claim.

Spanning of $H_d$ by ordered monomials. We show that every element of $H_d$ is a $\mathbb{C}$-linear combination of the images of ordered monomials $x^{r}w$; since words span ${\mathcal{F}}_d$ by [F4], it suffices to treat the image of a word $W$ in the symbols. Use strong induction on the pair $(n,\iota )$ ordered lexicographically, where $n$ is the number of $x$-symbols in $W$ and $\iota$ is the number of pairs of positions $(p<q)$ such that the $p$-th symbol is some $s_i$ and the $q$-th symbol is some $x_j$. If $n=0$, then $W$ is a word in the $s_i$; by the Coxeter presentation [F1] the subalgebra generated by the $s_i$ is spanned by the images of the elements $w\in S_d$, so the claim holds. If $n>0$ and every $x$-symbol precedes every $s$-symbol in $W$, then $W$ is of the form (word in $x$'s)(word in $s$'s), and the claim follows from commutativity of the $x$'s and the case $n=0$. Otherwise $W=W_1{s}_i{x}_j{W}_2$ with adjacent symbols; use $s_i{x}_{i+1}=x_i{s}_i+1$ if $j=i+1$, $s_i{x}_i=x_{i+1}{s}_i-1$ if $j=i$ (the relation $x_{i+1}{s}_i=s_i{x}_i+1$ solved for $s_i{x}_i$), and $s_i{x}_j=x_j{s}_i$ if $j\mathrm{\notin }\{i,i+1\}$. In every case $W\equiv W_1{x}_{j^{\mathrm{\prime }}}{s}_i{W}_2\pm W_1{W}_2$, where the first word has the same $n$ and one fewer $(s,x)$-inversion pair, while the second has $n-1$ $x$-symbols; both fall under the induction hypothesis. Hence the images of ordered monomials span $H_d$.

Jucys-Murphy elements. For $1\le j\le d$ put $J_j:={\sum }_{k<j}(kj)\in \mathbb{C}[S_d]$, so $J_1=0$. Then (i) $J_i{J}_j=J_j{J}_i$ for all $i,j$; (ii) $s_i{J}_j=J_j{s}_i$ for $j\mathrm{\notin }\{i,i+1\}$; and (iii) $s_i{J}_{i+1}=J_i{s}_i+1$. For (i) with $i<j$ write ${\mathrm{\Omega }}_m:={\sum }_{1\le k<l\le m}(kl)$ for the sum of all transpositions in $S_m$; ${\mathrm{\Omega }}_m$ is central in $\mathbb{C}[S_m]$ because conjugation permutes the transpositions, and $J_m={\mathrm{\Omega }}_m-{\mathrm{\Omega }}_{m-1}$. Since $J_i\in \mathbb{C}[S_i]\subseteq \mathbb{C}[S_{m-1}]\subset \mathbb{C}[S_m]$ for $i<m$, centrality of ${\mathrm{\Omega }}_m$ in $\mathbb{C}[S_m]$ and of ${\mathrm{\Omega }}_{m-1}$ in $\mathbb{C}[S_{m-1}]$ gives $[J_i,J_m]=[J_i,{\mathrm{\Omega }}_m]-[J_i,{\mathrm{\Omega }}_{m-1}]=0$. For (ii), conjugation by $s_i$ sends each transposition $(kj)$ to $(s_i(k)s_i(j))$; if $j<i$ both entries are fixed, and if $j>i+1$ the sum over $k<j$ contains the pairs with $k=i$ and $k=i+1$, whose images are interchanged, so the sum is preserved. For (iii), conjugation sends $(ki+1)$ to $(ki)$ for $k<i$ and fixes $(ii+1)$, so $s_i{J}_{i+1}{s}_i=J_i+s_i$; multiplying on the right by $s_i$ and using $s_i^2=1$ gives (iii).

The model satisfies the relations. Since ${\sigma }_i{m}_{a_i}=m_{{\sigma }_i(a_i)}{\sigma }_i=m_{-a_i}{\sigma }_i$, the operator identity $(1+{\sigma }_i)m_{a_i}=m_{a_i}(1-{\sigma }_i)$ and the rules ${\sigma }_i{\delta }_i=-{\delta }_i$, ${\delta }_i{\sigma }_i={\delta }_i$ hold; hence ${\delta }_i^2=m_{a_i}(1+{\sigma }_i)m_{a_i}(1+{\sigma }_i)=m_{a_i^2}(1-{\sigma }_i)(1+{\sigma }_i)=0$ and $T_i^2={\sigma }_i^2-{\sigma }_i{\delta }_i-{\delta }_i{\sigma }_i+{\delta }_i^2=1+{\delta }_i-{\delta }_i+0=1$. For $j\mathrm{\notin }\{i,i+1\}$ both ${\sigma }_i$ and ${\delta }_i$ commute with $m_{x_j}$, so $T_i{m}_{x_j}=m_{x_j}{T}_i$. For the cross relations, use ${\sigma }_i{m}_{x_{i+1}}=m_{x_i}{\sigma }_i$, ${\sigma }_i{m}_{x_i}=m_{x_{i+1}}{\sigma }_i$ and the two identities $m_{x_i}{\delta }_i-{\delta }_i{m}_{x_{i+1}}=1$ and ${\delta }_i{m}_{x_i}-m_{x_{i+1}}{\delta }_i=1$: for the first, $m_{x_i}{\delta }_i=m_{a_i{x}_i}(1+{\sigma }_i)$ while ${\delta }_i{m}_{x_{i+1}}=m_{a_i{x}_{i+1}}+m_{a_i{x}_i}{\sigma }_i$, and their difference is $m_{a_i(x_i-x_{i+1})}=1$; the second is the same computation with the roles of $x_i$ and $x_{i+1}$ interchanged. Therefore $T_i{m}_{x_{i+1}}-m_{x_i}{T}_i=(m_{x_i}{\sigma }_i-m_{x_i}{\sigma }_i)+(m_{x_i}{\delta }_i-{\delta }_i{m}_{x_{i+1}})=1$ and $m_{x_{i+1}}{T}_i-T_i{m}_{x_i}=1$. For $\mathrm{\mid }i-j\mathrm{\mid }>1$, the variables and swaps in $T_i,T_j$ are disjoint, so they commute. For the braid relation put $A=(x_i-x_{i+1}{)}^{-1}$, $B=(x_{i+1}-x_{i+2}{)}^{-1}$, $C=(x_i-x_{i+2}{)}^{-1}$, so $AB=C(A+B)$. The product rule of step 1.3 expands both $T_i{T}_{i+1}{T}_i$ and $T_{i+1}{T}_i{T}_{i+1}$ with coefficients, in the order $1,{\sigma }_i,{\sigma }_{i+1},{\sigma }_i{\sigma }_{i+1},{\sigma }_{i+1}{\sigma }_i,{\sigma }_i{\sigma }_{i+1}{\sigma }_i$, equal to $(-C(1+AB),BC(1-A),AC(1-B),-B(1-A)(1-C),-A(1-B)(1-C),(1-A)(1-B)(1-C))$. Here $A(B-C)=BC$ and $B(A-C)=AC$, both consequences of $AB=C(A+B)$, give the two shorter-swap coefficients and the common identity coefficient. Thus the braid relation holds. Hence the assignment $x_i\mapsto m_{x_i}$, $s_i\mapsto T_i$ preserves all the relations of the Definition, and by the universal properties [F4] it defines an algebra homomorphism $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:H_d\to {\mathrm{End}}_{\mathbb{C}}(\mathrm{\Omega })$.

Triangular expansion. Fix $w\in S_d$ with a reduced word $w=s_{i_1}\cdots s_{i_k}$, and put $c:={\prod }_{t=1}^k({\sigma }_{i_1}\cdots {\sigma }_{i_{t-1}})(1-a_{i_t})\mathrm{\ne }0$ in $\mathrm{\Omega }$, with the empty prefix acting as the identity. Every transported factor is nonzero because the prefix is a field automorphism. In the expansion of $T_w={\prod }_t((1-a_{i_t}){\sigma }_{i_t}-a_{i_t})$ by the product rule of step 1.3, the term choosing the factor $(1-a_{i_t}){\sigma }_{i_t}$ for every $t$ contributes $c{\sigma }_w$ after the scalar coefficients are moved left, while every other term is $m_g{\sigma }_v$ with $v$ the product of a proper subword of $s_{i_1}\cdots s_{i_k}$ in the induced order. If $\mathrm{\ell }$ denotes the minimal number of generators needed to express a permutation, then such a proper subword $J$ has $\mathrm{\ell }(v)\le \mathrm{\mid }J\mathrm{\mid }<k=\mathrm{\ell }(w)$, using subadditivity of $\mathrm{\ell }$ and reducedness. Collecting the expansion in the $\mathrm{\Omega }$-basis $\{{\sigma }_v\}$ therefore gives $T_w=c{\sigma }_w+{\sum }_{\mathrm{\ell }(v)<\mathrm{\ell }(w)}{c}_v{\sigma }_v$ with $c\mathrm{\ne }0$ and $c_v\in \mathrm{\Omega }$.

The evaluation homomorphism. Assume $d\ge 1$. The assignment $x_j\mapsto a+J_j$, $s_i\mapsto s_i$ defines an algebra homomorphism $\phi \text{\hspace{0.17em}⁣}:{\mathcal{F}}_d\to \mathbb{C}[S_d]$: the images of the $x$'s commute by step 1.5(i); the images of the $s_i$ satisfy the Coxeter relations by [F1]; and the cross relations follow from step 1.5(ii) and from $s_i(a+J_{i+1})=(a+J_i)s_i+1$, $(a+J_{i+1})s_i=s_i(a+J_i)+1$, which are step 1.5(iii) read in both directions. Since $\phi (x_1-a)=a+J_1-a=0$, the quotient universal property [F4] yields an algebra homomorphism $\bar{\phi }\text{\hspace{0.17em}⁣}:H_d\mathrm{/}(x_1-a)\to \mathbb{C}[S_d]$, which is surjective because the $s_i$ generate $S_d$.

Independence of the monomials in the model. By step 2.2 the change-of-basis matrix from $\{T_w\}$ to $\{{\sigma }_w\}$ inside $D$ is triangular with nonzero diagonal when permutations are ordered by length, hence invertible over the field $\mathrm{\Omega }$; thus $\{T_w:w\in S_d\}$ is an $\mathrm{\Omega }$-basis of $D$. Suppose ${\sum }_{r,w}{c}_{r,w}{m}_{x^r}{T}_w=0$ in $D$ with $c_{r,w}\in \mathbb{C}$ finitely supported. Grouping by $w$ gives ${\sum }_w{P}_w{T}_w=0$ with $P_w={\sum }_r{c}_{r,w}{x}^r\in \mathbb{C}[x_1,\dots ,x_d]\subseteq \mathrm{\Omega }$; $\mathrm{\Omega }$-independence of the $T_w$ forces $P_w=0$ for every $w$, and linear independence of the monomials [F3] then forces $c_{r,w}=0$. Hence the elements $\mathrm{\Phi }(x_1^{r_1}\cdots x_d^{r_d}w)$ are $\mathbb{C}$-linearly independent.

Basis theorem. Let $h\in H_d$ and write $h=\sum c_{r,w}{x}^{r}w$ using step 1.4. If $h=0$, applying the homomorphism $\mathrm{\Phi }$ of step 2.1 gives $\sum c_{r,w}{m}_{x^r}{T}_w=0$, and step 3.1 yields $c_{r,w}=0$. Hence the ordered monomials form a $\mathbb{C}$-basis of $H_d$, and the $\mathbb{C}$-linear map $\mathbb{C}[x_1,\dots ,x_d]{\otimes }_{\mathbb{C}}\mathbb{C}[S_d]\to H_d$, $f\otimes w\mapsto fw$, is an isomorphism by [F5]; in particular $\mathbb{C}[x_1,\dots ,x_d]$ and $\mathbb{C}[S_d]$ are subalgebras of $H_d$ and the cross relations hold. The elementary cases $H_0=\mathbb{C}$, $H_1=\mathbb{C}[x_1]$ were recorded in step 1.1.

The inverse map. Assume $d\ge 1$. In $H_d$ the relation $s_i{x}_{i+1}=x_i{s}_i+1$ gives, after left multiplication by $s_i$ and use of $s_i^2=1$, the recursion $x_{i+1}=s_i{x}_i{s}_i+s_i$. Hence in $H_d\mathrm{/}(x_1-a)$ one has $x_1\equiv a$ and inductively $x_j\equiv a+u_j$ with $u_j$ in the subalgebra generated by the images of the $s_i$, namely $u_1=0$ and $u_{j+1}=s_j{u}_j{s}_j+s_j$. By the basis theorem of step 4.1 every element of $H_d$ is a combination of monomials $x^{r}w$, and replacing each $x_j$ by $a+u_j$ expresses its image in $H_d\mathrm{/}(x_1-a)$ as an element of the image of the subalgebra generated by the $s_i$; by [F1] and [F2] that image is the $\mathbb{C}$-span of $\{w:w\in S_d\}$. Therefore the canonical map $\rho \text{\hspace{0.17em}⁣}:\mathbb{C}[S_d]\to H_d\mathrm{/}(x_1-a)$, $w\mapsto w+(x_1-a)$, has image equal to the whole quotient, and it is injective because $\bar{\phi }\circ \rho ={\mathrm{id}}_{\mathbb{C}[S_d]}$ by construction; hence $\rho$ is an isomorphism. Under it $x_j$ corresponds to $a+u_j$, and $\bar{\phi }(a+u_j)=a+J_j$ by step 2.3, so $\bar{\phi }(x_j)=a+{\sum }_{k<j}(kj)$; in particular $x_1$ corresponds to the scalar $a$, while for $j\ge 2$ the image is not the scalar $a$. The specified images of both the $x_j$ and the $s_i$ determine the homomorphism uniquely because these elements generate the quotient.

The degree filtration. For $k\ge 0$ let $F_k\subseteq H_d$ be the image of the $\mathbb{C}$-span of all words in the symbols containing at most $k$ of the symbols $x_1,\dots ,x_d$. Steps 1.4 and 4.1 identify $F_k$ with the span of the monomials $x^{r}w$ with $\mathrm{\mid }r\mathrm{\mid }\le k$. The substitution rules of step 1.4 replace an adjacent pair $s_i{x}_j$ by terms containing at most one $x$-symbol each, so a product of a word with $k$ $x$-symbols and a word with $m$ $x$-symbols rewrites into words with at most $k+m$ $x$-symbols; hence $F_k{F}_m\subseteq F_{k+m}$ and $(F_k{)}_{k\ge 0}$ is an algebra filtration with $1\in F_0$.

The associated graded is a semidirect product. Let $S:=\mathbb{C}[x_1,\dots ,x_d]⋊\mathbb{C}[S_d]$ be the algebra with $\mathbb{C}$-basis $x^{r}w$ and product $x^{r}w\cdot x^{r^{\mathrm{\prime }}}{w}^{\mathrm{\prime }}=x^{r+w\cdot r^{\mathrm{\prime }}}w{w}^{\mathrm{\prime }}$, where $(w\cdot r^{\mathrm{\prime }}{)}_j=r_{w^{-1}(j)}^{\mathrm{\prime }}$; equivalently, $S$ is generated by the symbols $x_i$ and $w$ subject to the relations obtained from those of $H_d$ by deleting the inhomogeneous term $1$ from both cross relations, i.e. $s_i{x}_{i+1}=x_i{s}_i$ and $x_{i+1}{s}_i=s_i{x}_i$. In $\mathrm{g}\mathrm{r}{H}_d$ the defining relations of $H_d$ project to these relations: the commutativity, Coxeter and distant-commutation relations are homogeneous of the stated degrees, while the two cross relations have the form (degree $1$ element) $=1$ (degree $0$ element), so their degree-$1$ parts are exactly $s_i{x}_{i+1}=x_i{s}_i$ and $x_{i+1}{s}_i=s_i{x}_i$. Consequently there is a surjective graded algebra homomorphism $S\to \mathrm{g}\mathrm{r}{H}_d$ sending $x_i$ and $w$ to the symbols of $x_i$ and $w$. In degree $k$ both sides are $\mathbb{C}$-spanned by the symbols $x^{r}w$ with $\mathrm{\mid }r\mathrm{\mid }=k$: these form a spanning set of ${\mathrm{g}\mathrm{r}}_k{H}_d$ by step 5.2 together with the basis theorem of step 4.1, and they form the basis of the degree-$k$ part of $S$ by construction; a surjection between $\mathbb{C}$-vector spaces with a spanning set of the same finite cardinality as a basis of the target is an isomorphism. Hence $\mathrm{g}\mathrm{r}{H}_d\cong S$ in every degree, and the associated graded is the semidirect product $\mathbb{C}[x_1,\dots ,x_d]⋊\mathbb{C}[S_d]$ in which $s_i$ swaps $x_i$ and $x_{i+1}$, as claimed.

Polynomial grading boundaries. For $d=0$, step 1.1 gives $H_0=\mathbb{C}$ with its degree-zero grading; for $d=1$, it gives $H_1=\mathbb{C}[x_1]$ with the usual polynomial grading. For $d\ge 2$, suppose there were an algebra grading $H_d={⨁}_{k\ge 0}{H}_d^{(k)}$ with $x_i$ in degree $1$ and $s_i$ in degree $0$. Then the relation $s_i{x}_{i+1}-x_i{s}_i=1$ would have its left-hand side in degree $1$ and its right-hand side in degree $0$, forcing $1=0$, which is impossible. Hence for $d\ge 2$ the polynomial-degree filtration of step 5.2 does not arise from that generator grading; its cross relation is inhomogeneous.