Rational Cherednik Algebras and Their Category O

For $r>0$, let $P_r$ be parallel transport along the arc $z(t)=r{e}^{it}$, $0\le t\le \theta$, and put $M_r:=g{P}_r$, an endomorphism of the initial fiber. The connection form $\mathrm{\Omega }=A(z)dz\mathrm{/}z$ is flat on the punctured disk: holomorphicity gives $d\mathrm{\Omega }=0$, and $\mathrm{\Omega }\wedge \mathrm{\Omega }=0$ because both factors are multiples of $dz$. Hence parallel transport around the boundary of an annular sector is the identity. If $r_1,r_2$ are two small radii, radial parallel transport identifies the initial fibers and the endpoint fibers; the coefficient identity in [F2] makes the resulting square commute with the endpoint identifications. Thus the two $M_r$ are conjugate and have the same characteristic polynomial.

Since $A(r{e}^{it})\to R$ uniformly for $0\le t\le \theta$ as $r\to 0$, subtract the integral equations for $P_r(t)$ and $e^{itR}$ from [F1]. The difference norm is bounded by ${\int }_0^t\mathrm{\parallel }A(r{e}^{iu})-R\mathrm{\parallel }\mathrm{\parallel }{P}_r(u)\mathrm{\parallel }du+{\int }_0^t\mathrm{\parallel }R\mathrm{\parallel }\mathrm{\parallel }{P}_r(u)-e^{iuR}\mathrm{\parallel }du$; Grönwall's inequality gives $P_r(\theta )\to e^{i\theta R}$. Therefore $M_r\to g{e}^{i\theta R}$.

The characteristic polynomial is a polynomial in matrix entries, hence continuous. Step 1.1 makes it constant for $r>0$, so its common value equals the characteristic polynomial of the limit in step 1.2, namely that of $g{e}^{i\theta R}$.

Under $w\in W$, the summand indexed by $s$ in $\kappa (wu,wv)$ is $-c_s{\omega }_s(wu,wv)s$. Reindex by $r=ws{w}^{-1}$. Conjugacy invariance gives $c_r=c_s$, and the reflected form satisfies ${\omega }_r(wu,wv)={\omega }_s(u,v)$ by [F1]–[F2]. The identity term is invariant because $w$ preserves $\omega$. Thus $\kappa (wu,wv)=w\kappa (u,v)w^{-1}$.

The scalar term $t\omega (v,w)1$ commutes with $u$. For a fixed reflection $s$, the contribution to the cyclic Jacobi sum is $-c_s({\omega }_s(v,w)(1-s)u+{\omega }_s(w,u)(1-s)v+{\omega }_s(u,v)(1-s)w)s$ by [F3].

Project $u,v,w$ to $U_s$ along $\ker (1-s)$. On the two-dimensional space $U_s$, for any alternating form $\eta$ and vectors $a,b,d$, one has $\eta (b,d)a+\eta (d,a)b+\eta (a,b)d=0$: the left side is alternating trilinear in a two-dimensional space and therefore vanishes. Apply $1-s$ to this identity with $\eta ={\omega }_s$; because ${\omega }_s$ vanishes on the radical, the result is exactly the parenthesized vector in step 1.2. Each reflection contribution is zero, so the cyclic sum vanishes.

Put all $x_i$ before every group symbol and all group symbols before all $y_j$, and order the $x_i$ and $y_j$ within their respective blocks by index. Use the defining relations as reductions: sort inverted pairs of $x$'s and of $y$'s; replace $wx$ by $w(x)w$; replace $yw$ by $w{w}^{-1}(y)$; replace $yx$ by $xy+\kappa (y,x)$; and multiply adjacent group symbols. Each vector in $w(x)$ or $w^{-1}(y)$ is expanded in the chosen finite bases. The irreducible words for these rules are precisely the displayed ordered monomials.

The reductions terminate. Use the lexicographic measure (i) number of vector letters, (ii) number of inversions of letter types for $x<\text{group}<y$, ignoring basis indices, (iii) number of group symbols, and (iv) sum of index inversions among the $x$ letters and among the $y$ letters. A correction term $\kappa (y,x)$ strictly decreases (i). The main mixed swap and each group crossing strictly decrease (ii), although expanding the group action may change (iv). Multiplication of adjacent group symbols cannot increase (ii) and, when it preserves (ii), decreases (iii); delete an identity group symbol by the same measure. A same-side sorting rule preserves (i)–(iii) and strictly decreases (iv). Every reduction therefore decreases a well-founded measure on finite words, so no infinite reduction sequence exists.

All overlapping reductions have the following forms. Overlaps among two adjacent group multiplications or group-crossing rules agree because the action of $W$ is an action and group multiplication is associative. Overlaps among same-side sorting rules agree because the $x$'s commute with one another and the $y$'s commute with one another. The forks $w{x}_j{x}_i$ and $y_j{y}_{i}w$ also join: a linear action preserves these commutativity relations, so applying the action before or after sorting gives the same symmetric product. A mixed group overlap $ywx$ agrees along its two reductions exactly by $W$-equivariance of $\kappa$ in [F2]. The only remaining overlaps are triples of vector letters, with two $y$'s and one $x$, or one $y$ and two $x$'s; the difference between reducing the two adjacent mixed pairs is the cyclic Jacobi expression from [F2], hence is zero. Reductions at disjoint pairs commute by substitution. Thus every one-step fork joins.

Termination from step 2.1 and the joining property from step 2.2 imply unique normal forms: by induction on the well-founded measure, if two first reductions differ, their join has smaller measure and therefore has the same normal form along both paths. Hence every word reduces uniquely to a linear combination of irreducible words. The reduction rules are consequences of [F1], so these words span the quotient. Conversely, the normal-form map annihilates each defining relation and all its left and right multiples; if a linear combination of irreducible words belonged to the defining ideal, its normal form would be both that combination and zero, so every coefficient would vanish. The irreducible words are therefore linearly independent in the quotient.

The irreducible words identified in step 1.1 are exactly $x_1^{a_1}\cdots x_n^{a_n}w{y}_1^{b_1}\cdots y_n^{b_n}$. Step 3.1 proves they form a basis. The tensor-product map in the Statement sends the tensor-product monomial basis to this basis, so it is an isomorphism. No restriction on $t$ or $c$ was used beyond the data definitions.

Since $E$ is finite dimensional, a finite basis of $E$ generates $\mathbb{C}[{\mathfrak{h}}^{\ast }]\otimes E$ as a $\mathbb{C}[{\mathfrak{h}}^{\ast }]$-module. By [F2] this is ${\mathrm{\Delta }}_c(E)$, so the finite-generation condition in [F1] holds.

Give $\mathbb{C}[{\mathfrak{h}}^{\ast }]\otimes E$ its polynomial-degree filtration. For $y\in \mathfrak{h}$, commute $y$ past one factor $x\in {\mathfrak{h}}^{\ast }$ by $yx=xy+[y,x]$. The first term preserves the number of polynomial factors after applying $y$ to the remaining factors; the commutator term replaces that $x$ by an element of $\mathbb{C}W$, which preserves degree and acts on the remaining polynomial. Since $y$ kills $E$, induction on polynomial degree shows that $y$ maps degree at most $d$ into degree at most $d-1$.

A product of $N$ elements of $\mathfrak{h}$ therefore maps every polynomial of degree at most $d$ to zero when $N>d$. Each element of ${\mathrm{\Delta }}_c(E)$ is a finite sum of such polynomial-degree vectors, so a sufficiently high-degree element of the augmentation ideal $S(\mathfrak{h}{)}_{>0}$ kills it. This is the local-nilpotence condition in [F1]. Both conditions hold, hence ${\mathrm{\Delta }}_c(E)\in {\mathcal{O}}_c(W,\mathfrak{h})$.

Choose a basis $x_1,\dots ,x_n$ of ${\mathfrak{h}}^{\ast }$ and its dual $y_1,\dots ,y_n$. For $s\in S$, let ${\lambda }_s\mathrm{\ne }1$ be the eigenvalue of $s$ on ${\alpha }_s\in {\mathfrak{h}}^{\ast }$; then $s{\alpha }_s^{\vee }={\lambda }_s^{-1}{\alpha }_s^{\vee }$ and $sy=y+{\lambda }_s^{-1}{\alpha }_s(y){\alpha }_s^{\vee }$. Put $\mathrm{e}\mathrm{u}:={\sum }_i{x}_i{y}_i+\frac{n}{2}-{\sum }_{s\in S}{c}_{s}s$. Using [F1], $[{\sum }_i{x}_i{y}_i,x]=x-{\sum }_s{c}_s{\alpha }_s^{\vee }(x){\alpha }_{s}s$ and $[{\sum }_i{x}_i{y}_i,y]=-y+{\sum }_s{c}_s{\lambda }_s^{-1}{\alpha }_s(y){\alpha }_s^{\vee }s$. The commutators of $-{\sum }_s{c}_{s}s$ with $x$ and $y$ are respectively $+{\sum }_s{c}_s{\alpha }_s^{\vee }(x){\alpha }_{s}s$ and $-{\sum }_s{c}_s{\lambda }_s^{-1}{\alpha }_s(y){\alpha }_s^{\vee }s$. Hence $[\mathrm{e}\mathrm{u},x]=x$ and $[\mathrm{e}\mathrm{u},y]=-y$. Also $\mathrm{e}\mathrm{u}$ commutes with $W$, because ${\sum }_i{x}_i{y}_i$ is the basis-independent contraction and ${\sum }_s{c}_{s}s$ is central by [F4].

On the degree-zero copy of an irreducible $W$-module $E$ in ${\mathrm{\Delta }}_c(E)$, $\mathrm{e}\mathrm{u}$ acts by the scalar $a_E$ of the central element $\frac{n}{2}-{\sum }_s{c}_{s}s$; Schur's lemma applies because $E$ is irreducible over $\mathbb{C}$. By $[\mathrm{e}\mathrm{u},x]=x$, the degree-$d$ part of ${\mathrm{\Delta }}_c(E)$ is an eigenspace of eigenvalue $a_E+d$.

For any $v\in M$, [F3] makes $U:=S(\mathfrak{h})\mathbb{C}Wv$ finite dimensional and stable under $S(\mathfrak{h})$ and $W$. The commuting nilpotent operators from a basis of $\mathfrak{h}$ have a nonzero common kernel in every nonzero such $U$: successively take the kernel of each basis operator, which is nonzero because that operator is nilpotent and is preserved by the remaining commuting operators. Call this common kernel $K$. It is $W$-stable. Since $\mathfrak{h}$ kills $K$, PBW gives $H_{1,c}K=\mathbb{C}[{\mathfrak{h}}^{\ast }]K$ as a vector space. The finite-dimensional space $K$ is locally finite under the linear operator $\frac{n}{2}-{\sum }_s{c}_{s}s$, and $[\mathrm{e}\mathrm{u},x]=x$ shows every polynomial-degree piece of $\mathbb{C}[{\mathfrak{h}}^{\ast }]K$ is finite dimensional and $\mathrm{e}\mathrm{u}$-stable. Thus $\mathrm{e}\mathrm{u}$ acts locally finitely on $H_{1,c}K$.

Induct on $\dim U$ to show that $\mathrm{e}\mathrm{u}$ acts locally finitely on $v$. The zero-dimensional case is immediate. In the nonzero case, $K\mathrm{\ne }0$ from step 2.2, so the image $\bar{U}$ of $U$ in $M\mathrm{/}{H}_{1,c}K$ has smaller dimension. By induction some nonzero polynomial $p$ satisfies $p(\mathrm{e}\mathrm{u})\bar{v}=0$, hence $p(\mathrm{e}\mathrm{u})v\in H_{1,c}K$. Step 2.2 gives a nonzero polynomial $q$ with $q(\mathrm{e}\mathrm{u})p(\mathrm{e}\mathrm{u})v=0$, so the span of $\{{\mathrm{e}\mathrm{u}}^{j}v:j\ge 0\}$ is finite dimensional. This proves local finiteness for every $v$.

For irreducible $E$, the standard module ${\mathrm{\Delta }}_c(E)$ has a unique simple head. Every proper submodule is stable under $\mathrm{e}\mathrm{u}$ and therefore is a sum of its eigenspaces; it cannot meet the degree-zero copy of $E$, since $E$ is irreducible and generates ${\mathrm{\Delta }}_c(E)$. The sum of all proper submodules still misses degree zero, so it is proper and contains every proper submodule. Its quotient has no nonzero proper submodule, hence is the unique simple head.

Choose finite generators of $M$ over $\mathbb{C}[{\mathfrak{h}}^{\ast }]$. By step 3.1 each generator is a finite sum of generalized $\mathrm{e}\mathrm{u}$-eigenvectors; replace the generating list by these finitely many components, with generalized eigenvalues ${\lambda }_1,\dots ,{\lambda }_r$. The relation $[\mathrm{e}\mathrm{u},x]=x$ shifts a generalized eigenvalue by $1$ under multiplication by $x$, so all eigenvalues of $M$ lie in ${\bigcup }_i({\lambda }_i+{\mathbb{Z}}_{\ge 0})$. For any fixed $\lambda$, only finitely many pairs $(i,d)$ satisfy ${\lambda }_i+d=\lambda$, and each $S^d({\mathfrak{h}}^{\ast })$ is finite dimensional; therefore every generalized eigenspace $M[\lambda ]$ is finite dimensional.

Let $p(M)=\{m\in M:ym=0\text{ for every }y\in \mathfrak{h}\}$. It is $W$-stable, and $\mathrm{e}\mathrm{u}$ acts on it as the central element $z=\frac{n}{2}-{\sum }_s{c}_{s}s$. Every vector of a $W$-module lies in the finite-dimensional span of its $W$-orbit; averaging a projection over finite $W$ shows each such span is semisimple. Thus every vector of $p(M)$ is a sum of $z$-eigenvectors with eigenvalues among the finitely many scalars $a_E$ for irreducible $W$-modules $E$. By step 4.1 the corresponding generalized eigenspaces $M[a_E]$ are finite dimensional, so $p(M)\subseteq {\sum }_{E}M[a_E]$ is finite dimensional. If $M\mathrm{\ne }0$, choose an eigenvalue of minimal real part among the finite union of lower-bounded integer progressions containing the eigenvalues of $M$. A nonzero vector in its generalized eigenspace is killed by $\mathfrak{h}$, since each $y$ lowers the $\mathrm{e}\mathrm{u}$-eigenvalue by one. Thus $p(M)\mathrm{\ne }0$.

Every nonzero object $N\in \mathcal{O}$ contains a simple submodule. By [F5]–[F6], $N$ is Noetherian, because it is finite over $\mathbb{C}[{\mathfrak{h}}^{\ast }]$; the same is true of each submodule and quotient. By step 5.1, $p(N)$ is a nonzero finite-dimensional $W$-module. Every finite-dimensional $W$-module is semisimple: for a $W$-stable subspace, average any linear projection onto it over the finite group to obtain a $W$-equivariant projection, then induct on dimension. Choose an irreducible $W$-submodule $E\subseteq p(N)$ maximal among those occurring in $p(N)$ for the order $E<F$ when $a_F-a_E\in {\mathbb{Z}}_{>0}$. Frobenius reciprocity gives a nonzero map ${\mathrm{\Delta }}_c(E)\to N$ whose image is $L:=H_{1,c}E$. Every irreducible constituent $F$ of $p(L)$ has Euler eigenvalue $a_F$, while $L$ is a quotient of ${\mathrm{\Delta }}_c(E)$, whose eigenvalues are $a_E+d$ with $d\ge 0$ by step 2.1. If $d>0$, then $F$ also occurs in $p(N)$ and contradicts maximality of $E$. If $d=0$, that eigenspace in ${\mathrm{\Delta }}_c(E)$ is exactly its top copy of $E$, so the only constituent is $E$. Thus $p(L)=E$. Every nonzero submodule of $L$ has a nonzero singular vector by step 5.1; its singular space is a nonzero $W$-submodule of the irreducible $E$, so it contains $E$ and therefore equals $L$. Thus $L$ is simple. The same induction-reciprocity argument shows that any simple object is a quotient of some ${\mathrm{\Delta }}_c(E)$.

Starting with $M_0=0$, if $M\mathrm{/}{M}_j\mathrm{\ne }0$, choose a simple submodule of that quotient by step 6.1 and let $M_{j+1}$ be its inverse image in $M$. This gives a strictly increasing chain of submodules. Noetherianity from step 6.1 forces the chain to stop after finitely many steps, at $M$. Its successive quotients are simple, so it is a finite composition series. This proves the Statement.

For $f\in \mathbb{C}[{\mathfrak{h}}^{\ast }]$ and $s\in S$, the polynomial $f-sf$ vanishes on $\ker {\alpha }_s$, since $s^{-1}$ fixes every point of that hyperplane. The ideal of this hyperplane is generated by ${\alpha }_s$, so $(1-s)f\mathrm{/}{\alpha }_s$ is a polynomial. Thus every displayed $y$-operator preserves $\mathbb{C}[{\mathfrak{h}}^{\ast }]$.

Let $R_s={\alpha }_s^{-1}(1-s)$ on the rational function field of $\mathfrak{h}$. For multiplication by $x\in {\mathfrak{h}}^{\ast }$, $[R_s,x]={\alpha }_s^{-1}(x-sx)s={\alpha }_s^{\vee }(x)s$ by [F4]. Using $[{\mathrm{\partial }}_y,x]=x(y)$ therefore gives $[{\mathrm{\partial }}_y-{\sum }_s{c}_s{\alpha }_s(y)R_s,x]=x(y)1-{\sum }_s{c}_s{\alpha }_s(y){\alpha }_s^{\vee }(x)s$, exactly [F1]. The $x$'s commute, and conjugation by $w$ sends the operator for $y$ to that for $wy$ because $c$ and the reflection forms are equivariant.

Write $T_y={\mathrm{\partial }}_y-{\sum }_s{c}_s{\alpha }_s(y)R_s$. The commutator $[T_y,T_z]$ is a differential-reflection operator on ${\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}}$, the complement of the reflection hyperplanes. By step 1.2 and Jacobi, ${\mathrm{ad}}_{[T_y,T_z]}(x)=[T_y,\kappa (z,x)]-[T_z,\kappa (y,x)]=[T_y,\kappa (z,x)]+[T_z,\kappa (x,y)]=0$, since $\kappa (y,z)=0$ for $y,z\in \mathfrak{h}$ and [F2] applies. The commutator with a group element uses the same formula as in [F1], by the $W$-covariance just checked.

In the crossed differential-operator algebra $\mathcal{D}({\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}})⋊W={⨁}_{w\in W}\mathcal{D}({\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}})w$, the centralizer of all multiplication operators by $x\in {\mathfrak{h}}^{\ast }$ consists of multiplication by rational functions. Indeed, for a nonidentity $w$, choose $x$ with $wx\mathrm{\ne }x$; the top differential-order term of $[Dw,x]$ is the top term of $D$ multiplied by $wx-x$, so commutation forces $D=0$. For $w=1$, a differential operator commuting with every coordinate function has order zero, since the commutator of a positive-order operator with a coordinate has nonzero top symbol in some coordinate direction. Thus [T_y,T_z], which commutes with all $x$ by step 2.1, is multiplication by a rational function. Every $T_y$ kills $1$, so $[T_y,T_z]1=0$, and that multiplication function is zero. Hence the $T_y$ commute.

The assignments $x\mapsto m_x$, $w\mapsto w$, and $y\mapsto T_y$ now satisfy every relation in [F1]: the $x$'s commute, the $y$'s commute by step 3.1, the mixed commutator is step 1.2, and the $W$-action relation is the covariance in step 1.2. The quotient universal property therefore gives an $H_{1,c}$-module structure on $\mathbb{C}[{\mathfrak{h}}^{\ast }]$.

The constant polynomial is killed by every $T_y$, and $W$ fixes it, so induction gives an $H_{1,c}$-module map $P_c={\mathrm{\Delta }}_c(\mathbf{1})\to \mathbb{C}[{\mathfrak{h}}^{\ast }]$ sending $1\otimes 1$ to $1$. It is surjective because multiplication by ${\mathfrak{h}}^{\ast }$ generates all polynomials. By PBW [F3], both spaces have the same monomial basis and this map is the identity on that basis, hence is an isomorphism. This is the asserted polynomial representation and action.