Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras
aa r X i v : . [ m a t h . QA ] J u l TRACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMALCHEREDNIK ALGEBRAS
ALEXANDER VITANOVAbstract.
Etingof introduces a sheaf of Cherednik algebras ℋ ,c,X,G , attached to a complex algebraic variety X with an action by a finite group G , by means of generators and relations. In a previous paper the author showedhow to realize the sheaf of Cherednik algebras on a general global quotient orbifold X/G by gluing sheaves of flatsections of flat holomorphic vector bundles on orbit type strata in X . In the current note we use this realizationof the sheaf of global Cherednik algebras to define trace density maps for ℋ ,c,X,G attached to the various orbittype strata. The maps are not entirely explicit. By means of the trace density morphisms we obtain a morphismbetween the hypercohomology of the sheaf of Hochschild chain complexes of ℋ ,c,X,G for a Coxeter group G and the Chen-Ruan orbifold cohomology of X/G . We subsequently show that for any complex reflection groupG, when c is a formal parameter, the hyercohomology of the sheaf of Hochschild chain complexes of ℋ ,c,X,G is isomorphic to the Chen-Ruan cohomology of X/G with values in the Laurent field. Finally, we present analgebraic index theorem for the sheaf of formal global Cherendik algebras ℋ , (( ~ )) ,X,G localized at ~ . Contents
1. Introduction 12. Preliminaries 22.1. Complex reflections in linear spaces 22.2. Rational Cherednik algbera 42.3. Global Cherednik algebra 53. Hochschild and cyclic homology of some algebras 64. Trace densities and hypercohomology 144.1. Hochschild n − l -cocycle of the Harish-Chandra modules 𝒜 Hn − l,l and 𝒜 H, (( ~ )) n − l,l
1. Introduction
In [Vit19] it was shown by virtue of Gelfand-Kazhdan formal geometry how to realize the sheaf of Chered-nik algebras ℋ ,c,X,G in terms of gluing of sheaves of flat sections of flat holomorphic bundles which aredefined on the orbit type strata of X and arise from special Harish-Chandra torsors with fibers associativealgebras with a Harish-Chandra module structure. By means of the constriction in [Vit19] one can uniquely Department of Mathematics, MIT
E-mail address : [email protected] . represent every section of ℋ ,c,X,G over any G -invariant open set U in X as a string of flat sections of theholomorphic bundles on those orbit type strata intersecting U . In the current note we invoke this identifica-tion to define for every orbit type stratum X iH a morphism from the Hochschild chain complex of ℋ ,c,X,G tothe complex of de Rham differential forms, shifted by a degree of codimension, on the connected fixed pointsubmanifold component X Hi containing X iH . These morphisms are referred to as trace density morphism orsimply trace densities because their images are de Rham cohomology classes which in turn can be integratedover compact submanifolds to singular cohomology classes in H −• ( X Hi , C )[2 n − l ] where l = codim( X Hi ) .In particular, one obtains this way non-trivial trace maps for the sheaf of Cherednik algebras, which can beseen as a sheaf-theoretic generalization of the linear traces for the rational Cherednik algebra constructedby Etingof, Ginzburg and Berest in [BEG03a; BEG03b; BEG04]. When X is a compact manifold, we obtain acomplete list of explicit formulas for the trace maps of the global formal Cherednik algebra H , (( ~ )) ( U, G ) , tr : HH ( H , (( ~ )) ( U, G )) → H n − l ( X Hi , C ) ∼ = C for all X Hi intersecting U . We employ the trace density morphisms to define morphisms from the hypercoho-mology of the sheaf of Hochschild chain complexes of ℋ ,c,X,G , respectively of ℋ , (( ~ )) ,X,G to the Chen-Ruancohomology of X/G . We demonstrate that in the formal case, the corresponding morphism is in fact an iso-morphism generalizing a result from [AFLS00]. We remark that in a separate closely related work [Vit20]we adapt the constructions presented here to the holomorphic and cohomological settings and prove sub-sequently with their help that the sheaf of twisted formal Cherednik algebras is a universal formal filtereddeformation of 𝒟 X ⋊ G - until recently an open problem which has resisted a rigorous proof for over a decade.Finally, mimicking techniques in [FFS05; EF08], [PPT07] and [RT12] we prove an algebraic index theorem forthe trace density morphisms χ Hi, (( ~ )) which provide a value of the induced linear traces for the sheaves ofCherednik algebras at id . We find out that the obtained trace maps distinguish between 𝒟 X ⋊ G and itsformal deformation. This is in a stark contrast to the Euler characteristics which do not distinguish betweenthe sheaf of formal Cherednik algebras and 𝒟 X ⋊ G .
2. Preliminaries
Complex reflections in linear spaces.
Let h be a finite-dimensional complex vector space and let h ∗ bethe dual space of h . A semisimple endomorphism s of h is called a complex reflection in h if rank(id h − s ) = 1 .The fixed point subspace h s := ker(id h − s ) of the complex reflection s ∈ End( h ) is a hyperplane, which isreferred to as the reflecting hyperplane of s . Suppose G is a finite subgroup of GL( h ) ⊂ End( h ) and let 𝒮 denote the set of all complex reflections in h contained in G . The group G is said to be a complex reflectiongroup if it is generated by 𝒮 . Since G is finite, all complex reflections s ∈ 𝒮 have a finite order which ingeneral might be higher than two. The group G acts naturally on h ∗ via the dual representation defined by ( g · ω )( v ) := ω ( g − · v ) for all g ∈ G , ω ∈ h ∗ , v ∈ h and hence every complex reflection s ∈ G defines aunique complex reflection s ∗ in h ∗ . For brevity we shall not distinguish between a given complex reflectionin h and its unique induced complex reflection in h ∗ and shall use the same notation s ∈ G for both.Given a complex reflection s ∈ 𝒮 in h we denote its unique non-trivial eigenvalue by λ ∨ s and by α ∨ s ∈ h aneigenvector of s in h corresponding to λ ∨ s which we call a root . On the one side, the root α ∨ s generates theimage of id h − s and on the other side it vanishes everywhere on the reflecting hyperplane of s in h ∗ , thatis, Im(id h − s ) = C α ∨ s and ker(id h ∗ − s ) = ker( α ∨ s ) . Similarly, we designate by λ s the unique non-trivialeigenvalue of s in h ∗ and by α s ∈ h ∗ an eigenvector of s in h ∗ corresponding to λ s , which we call coroot . RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 3
In analogous manner to α ∨ s the linear form α s generates the image of id h ∗ − s and vanishes identically onthe hyperplane of s in h , that is, Im(id h ∗ − s ) = C α s and ker(id h − s ) = ker( α s ) , respectively. Since theorder of any complex reflection s ∈ 𝒮 is finite, the corresponding eigenvalues λ ∨ s and λ s are (not necessarilyprimitive) roots of unity. We shall denote by α ∨ s ∈ h and α s ∈ h ∗ the unique eigenvectors of a given complexreflection s ∈ 𝒮 in h and in h ∗ , respectively, corresponding to the eigenvalues λ ∨ s and λ s , respectively.2.1.1. Irreducible well-generated complex reflection groups.
If a complex reflection group G ⊂ GL( h ) is suchthat h is an irreducible G -module, we call G an irreducible complex reflection group . The following theorem,which the reader can find in e.g. [LT09], shows that the study of complex reflection groups reduces to thestudy of irreducible complex reflection groups. We formulate the theorem in a slightly more general mannerthan stated in [LT09] which suits the purposes of this thesis better. Theorem 2.1 (Theorem 1.27, [LT09]).
Suppose that G is a finite complex reflection group on h . Then h is thedirect sum of subspaces h , h . . . , h m such that the subgroup G i of G , generated by complex reflections whoseroots lie in h i , acts irreducibly on h i for every i = 1 , . . . , m , and G ∼ = G × G × · · · × G m . If u is an irreducible G -submodule of h that is not fixed pointwise by every element of G , then u = h i for some i . It follows from this theorem that h = h G ⊕ h ⊕ · · · ⊕ h k , where the h i are the non-trivial irreducible G -modules. The support of a complex reflection group G ⊂ GL( h ) , denoted supp( G ) , is the algebraic com-plement of the subspace h G . Lemma 2.2.
The support of a complex reflection group G ⊂ GL( h ) is spanned by the roots of the complexreflections in G .Proof. Theorem 2.1 yields h = h G ⊕ supp( G ) . Take a vector v ∈ supp( G ) . Then there exists at least oneelement g ∈ G such that g = s . . . s r for some complex reflection generators s , . . . , s r ∈ S , and v / ∈ h g . Itfollows that v ∈ Im(1 − g ) . Then for some x ∈ h we have that v = (1 − s . . . s r ) x = x − s . . . s r x = (1 − s r ) x + s r x − s . . . s r x = (1 − s r ) x + (1 − s r − )( s r x ) + · · · + (1 − s )( s . . . s r x )= λ r α ∨ s r + λ r − α ∨ s r − + · · · + λ α ∨ s , where α ∨ s i is the root of s i for i = 1 , . . . , r . Hence, v ∈ span S { α ∨ s } and consequently supp( G ) ⊂ span S α ∨ s .Conversely, if v ∈ span S { α ∨ s } , then v ∈ supp G by the fact that by default no root α ∨ s lies in h G . Thus, span S α ∨ s ⊂ supp G (cid:3) The rank of a reflection group G , denoted rank( G ) , is the dimension of its support. If G has a generatingset S of complex reflections whose cardinality is equal to the rank of G , then G is called a well-generated complex reflection group. Irreducible well-generated complex reflection groups G are of particular interestto our work since they admit so called Coxeter elements . Let S ∗ denote the set of hyperplanes in h fixed bysome complex reflection. Set N := | S | and N ∗ = | S ∗ | . The Coxeter number of G is the constant h := N + N ∗ dim h .A ζ -regular element in G is a group element g ∈ G with eigenvalue ζ ∈ C such that no eigenvector v ∈ h of g is contained in a hyperplane in S ∗ . A Coxeter element is a ζ h -regular element in G where ζ h is a primitive h -th root of unity. Coxeter elements satisfy various nice properties. For the purposes of our work we want to TRACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS know the eigenvalues of a Coxeter element. The succeeding lemma, whose proof is based on various claimsmade in chapter in [Kan13], demonstrates that a Coxeter element has no eigenvalues equal to one. Lemma 2.3. If c is a Coxeter element in an irreducible well-generated complex reflection group G ⊂ GL( h ) witha Coxeter number h ∈ Z ≥ , c is a semisimple element and every eigenvalue of c is of the form ζ kh , ≤ k < h ,where ζ h ∈ C is a primitive h -th order of unity.Proof. The definition of a Coxeter element implies the existence of an eigenvector x of c in h whose isotropygroup Stab( x ) does not contain any complex reflections. On the other hand, Steinberg’s fixed point theoremstipulates that Stab( x ) is a complex reflection subgroup. Hence, Stab( x ) = { } . Thus the condition c h x = ζ h x = x implies that c is of order h . It is a standart fact from linear algebra that endomorphisms of finiteorder are diagonalizable over the comlex field, so c is diagonalizable. Furthermore, every eigenvalue of c is a h -th order of unity. Also by definition we know that the Coxeter element c possesses a primitive h -th root ofunity ζ h . Thus, every eigenvalue of c is of the form ζ kh for ≤ k < h , which concludes the proof. (cid:3) For the existence of a trace map on b H , (( ~ )) ( h , G ) later in this work it is crucial to make sure that everyirreducible well-generated complex reflection group possess at least one Coxeter element. We do just that inthe next lemma. Lemma 2.4.
An irreducible well-generated complex reflection group G has a Coxeter element.Proof. According to [Kan13, Corollary - A ] for a primitive d -th root of unity ζ there exists an element c ∈ G having ζ as an eigenvalue if and only if d divides a degree d i for some i = 1 , . . . , dim h . Let ζ h be aprimitive h -th root of unity. Since by default h = d dim h , there is an element c ∈ G having ζ h as an eigenvalue.Finally, [RRS17, Theorem . ] implies that c does not belong to S ∗ . Hence c is a Coxeter element. (cid:3) Rational Cherednik algbera.
To the data h , h ∗ and G one attaches the rational Cherednik algebra H t,c ( h , G ) : Definition 2.5.
The rational Cherednik algebra H t,c ( h , G ) is the quotient of the smash-product algebra T • ( h ⊗ h ∗ ) ⋊ C G by the ideal generated through gyg − − g y, gug − − g u, [ y, y ′ ] , [ u, u ′ ] , [ u, y ] − t ( u, y ) − X s ∈ 𝒮 c ( s ) ( u, α s ) ( y, α ∨ s ) s, where u, u ′ ∈ h and y, y ′ ∈ h ∗ and c ∈ C [ 𝒮 ] Ad G , where Ad refers to the adjoint action of G on itself and t ∈ C . The algebra H t,c ( h , G ) comes with two natural increasing filtrations . The first one is the Bernstein fitration e F • which is given by putting deg( G ) = 0 and deg( h ∗ ) = deg( h ) = 1 . The second one is the geometricfiltration F • , given by the rule deg( h ∗ ) = deg( G ) = 0 , deg( h ) = 1 . Let m be a maximal ideal in C [ h ] .The degree-wise completion b H t,c ( h , G ) of H t,c ( h , G ) as a C [ h ] -module is by definition the scalar extension of H t,c ( h , G ) to formal functions on the formal neighborhood of zero in h , that is, b H t,c ( h , G ) := C [[ h ]] ⊗ C [ h ] H t,c ( h , G ) . We remark that since the underlying C [ h ] -module of the rational Cherednik algebra is not finitelygenerated over the Noetherian ring C [ h ] , the degree-wise formal completion does not coincide with the formalcompletion of the Cherednik algebra as a C [ h ] -module with respect to the induced m -adic topology on the C [ h ] -module H t,c ( h , G ) , that is, the equality lim ↼i H t,c ( h , G ) / m i H t,c ( h , G ) ∼ = b H t,c ( h , G ) does not holdtrue. Nevertheless, for brevity we shall abuse notation denoting the degree-wise completion of the rational RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 5
Cherednik algebra by b H t,c ( h , G ) . b H t,c ( h , G ) inherits both increasing filtration e F • and F • from H t,c ( h , G ) bythe rule e F i b H t,c ( h , G ) := C [[ h ]] ⊗ C [ h ] e F i H t,c ( h , G ) , respectively F i b H t,c ( h , G ) := C [[ h ]] ⊗ C [ h ] F i H t,c ( h , G ) .The following theorem on the Hochschild homology and cohomology of H , ( h , G ) = 𝒟 ( h ) ⋊ G is a specialcase of [AFLS00, Theorem 6.1]: Theorem 2.6. (1) HH j ( 𝒟 ( h ) ⋊ G, 𝒟 ( h ) ⋊ G ) ∼ = HH n − j ( 𝒟 ( h ) ⋊ G, 𝒟 ( h ) ⋊ G ) ∼ = C a j , where a j is the number of conjugacy classes of elements in G having eigenvalue with multiplicity j . Global Cherednik algebra.
Assume that X is a n -dimensional complex manifold equipped with anaction by a finite group G of holomorphic automorphisms of X . Let X g ⊂ X denote the fixed point setof g ∈ G . A nonlinear complex reflection of X is a pair ( g, Y g ) consisting of a group element g ∈ G and aconnected component Y g of X g of complex codimension in X . A codimension connected component Y g ⊂ X g will be referred to throughout the text as a reflection hypersurface in accord with the terminologyestablished in [Eti04].We define the analogue of Dunkl-Opdam operators for complex reflection representations in the case of acomplex manifold with a finite group action. Let us denote by S the set of all complex reflections of X and let c : S −→ C be a G -invariant function. Let D := S ( g,Y g ) ∈ S Y g and let j : X \ D −→ X be theopen inclusion map. For each complex reflection ( g, Y g ) ∈ S let 𝒪 X ( Y g ) designate the sheaf of holomorphicfunctions on X \ Y g taking poles of at most first order only along Y g , let ξ Y g : 𝒯 X → 𝒪 X ( Y g ) / 𝒪 X be thenatural surjective map of 𝒪 X -modules, and let p : X −→ X/G denote the projection. A
Dunkl operatorassociated to a holomorphic vector field V on X is a section D V of the sheaf p ∗ j ∗ j ∗ ( 𝒟 X ⋊ G ) over an opensubset U ⊂ X/G , which in a G -invariant coordinate chart U ′ ⊂ p − ( U ) ⊂ X has the form(2) D V = ℒ V + X ( g,Y g ) ∈ S c (( g, Y g ))1 − λ ( g,Y g ) f Y g ( g − . Here ℒ V is the Lie derivative with respect to V , λ ( g,Y g ) is the nontrivial eigenvalue of g on the conormal bun-dle to Y g and f Y g ∈ Γ( U ′ , 𝒪 X ( Y g )) is a function whose residue agrees with with V once both are restricted tothe normal bundle of Y g in X , that is f Y g ∈ ξ Y g ( V ) . On each intersection U ′ ij := U ′ i ∩ U ′ j of G -invariant coor-dinate charts U ′ i ⊂ U and U ′ j ⊂ U the coordinate representations of D V | U ′ i and D V | U ′ j coincide, from whichit follows that D V is uniquely determined by its local coordinate representation in an arbitrary G -invariantchart. To the date X and G one attaches the following sheaf of non-commutative associative algebras: Definition 2.7.
The sheaf of Cherednik algebras ℋ ,c,X,G on the orbifold X/G is a subsheaf of the sheaf p ∗ j ∗ j ∗ ( 𝒟 X ⋊ G ) generated locally by p ∗ 𝒪 X , C G and Dunkl operators D V associated to holomorphic vectorfields V on X . The definition of H ,c,X,G is independent on the choice of a function f y g ∈ Γ( U ′ , 𝒪 X ( Y g )) in the Dunkloperators, because by adding a holomorphic function from Γ( U ′ , 𝒪 X ) to f Y g the new Dunkl operator differsfrom the old one by a section of 𝒪 X ⋊ G . The sheaf of Cherednik algebras ℋ ,c,X,G possesses a naturalincreasing and exhaustive filtration ℱ • which is defined on the generators by deg( 𝒪 X ) = deg( C G ) = 0 and deg( D V ) = 1 for Dunkl operators D V , V a holomorphic vector field on X . It is the analogue of the geometricfiltration of the rational Cherednik algebra, discussed in 2.2. We can equivalently define ℋ ,c,X,G in the so TRACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS called G -equivariant topology T GX of X . It is by definition comprised of the preimages of open sets in the quo-tient topology of X/G . Since the projection p : X → X/G is surjective, pp − ( U ) = U for any set U in X/G .This means in particular that every open set U in the orbifold is the image of some open set V in X under p whence the G -equivariant topology T GX of X consists of subsets of the form p − ( p ( V )) = S g ∈ G gV where V is an arbitrary open set in X . We note that the main difference of that topology as opposed to the standardone is that the G -equivariant topology is not Hausdorff in X . Next, we define a basis for the G -equivarianttopology T GX on X . Recall that a slice at a point x ∈ X is a Stab( x ) -invariant neighborhood W x such that W x ∩ gW x = ∅ for all g ∈ G \ Stab( x ) . A slice W x at a point x with Stab( x ) = H cannot contain anyelement y whose stabilizer Stab( y ) is not contained in H since the contrary would imply that there is a groupelement g ∈ G \ Stab( x ) , for which W x ∩ gW x = ∅ , which would then contradict the assumption, that W x isa slice. This implies that all strata, crossing a H -invariant slice W x , have an isotropy type contained in H . Aslice is called a linear if there is a Stab( x ) -invariant open set V in C n such that W x is Stab( x ) -equivariantlybiholomorphic to V . Since we assume that the group G is finite, it acts on X properly discontinuously andthus each point x in X possesses a slice W x . By Cartan’s Lemma one can shrink the slice W x till the Stab( x ) -action is linearized which implies that the G -action is locally linear. It is of crucial importance for our workthat a linear slice W x at a point x on a stratum X iH intersects apart from X iH only strata X jK of codimensionlower than the codimension of X iH such that x lies in the intersection of their closures which by the frontiercondition of the stratification means that X iH lies within T { X jK | X jK ∩ W x = ∅ } . For any given open set V in X and every x ∈ V with Stab( x ) := H ≤ G , there is an H -invariant linear slice W x ⊂ V which is H -equivariantly biholomorphic to a box in C n − l × C l where C n − l is the fixed point subspace of C n withrespect to H . Consequently, ind GH ( W x ) := ` g ∈ G/H gW x is a subset of S g ∈ G gV . Hence, the collection B GX of sets ind GH ( W x ) where W x is either an H -invariant linear slice biholomorphic to a box in C n − l × C l or an H -invariant open subset of ˚ X with gW x ∩ W x = ∅ for all g ∈ G/H , forms a basis for T GX .Note that for each x ∈ X Hi , every linear slice W x constitutes a holomorphic slice chart for the fixed pointsubmanifold X Hi . That is, if x , x , . . . , x n − l are the holomorphic coordinates on the complex vector sub-space (cid:0) C n (cid:1) H = C n − l and y , . . . , y l are the holomorphic coordinates on the l -dimensional complementof C n − l in C n , then ( x , . . . , x n − l , y , . . . , y l ) define local holomorphic coordiantes of X on W x such that ( x , . . . , x n − l ) are local holomorphic coordinates of W x ∩ X Hi and ( y , . . . y l ) are local holomorphic coordi-nates on W x in transversal direction to X Hi .
3. Hochschild and cyclic homology of some algebras
We start by revisiting the Hochschild and cyclic homology of some important for our applications specialalgebras. The theory of Hochschild and cyclic homology is a well-established field of homological algebra.The literature on that topic is vast whence we shall refrain from repeating the well-known definitions andfacts. Instead, we refer the reader to [Wei94] and [Lod13] for a detailed discussion thereof.Throughout the chapter let 𝒟 alg. ( h ) be the algebra of differential operators on h with algebraic coefficientsand let b 𝒟 ( h ) ⋊ G denote the degreewise completion of the smash-product algebra 𝒟 ( h ) alg. ⋊ G with respectto the m -adic topology on the ring C [ h ] . Proposition 3.1. (3) HH • ( b 𝒟 ( h ) ⋊ G ) ∼ = HH • ( 𝒟 alg. ( h ) ⋊ G ) RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 7
Proof.
Let us denote by C • := ( C • ( 𝒟 alg. ( h ) ⋊ G ) , d ) the Hochschild chain complex of 𝒟 alg. ( h ) ⋊ G withHochschild differential d and by b C • := ( b C • ( b 𝒟 ( h ) ⋊ G ) , d ) the completion of the Hochschild chain complexof the completed Weyl algebra b 𝒟 ( h ) ⋊ G . The Lie group U(1) acts on h by scalar multiplication and naturallyextends to an action by C -algebra automorphisms ρ : U(1) −→ Aut( b C n ) on b C n for every n ∈ Z ≥ by therule ρ ( λ )( a ⊗ · · · ⊗ a n ) := λ ∗ ◦ a ◦ λ − ∗ ⊗ · · · ⊗ λ ∗ ◦ a n ◦ λ − ∗ , where λ ∗ denotes the pullback action of U(1) on C [ h ] and ◦ is the composition in 𝒟 alg. ( h ) ⋊ G . This U(1) -action on b C n can be further extended to an action by graded algebra automorphisms on the chain complex b C • , if we understand b C • as a graded C -algebra. Furthermore, the computation d n ( ρ ( λ )( a ⊗ · · · ⊗ a n )) = d n ( λ ∗ ◦ a ◦ λ − ∗ ⊗ · · · ⊗ λ ∗ ◦ a n ◦ λ − ∗ )= n X j =0 ( − j λ ∗ ◦ a ◦ λ − ∗ ⊗ · · · ⊗ λ ∗ ◦ a j a j +1 ⊗ · · · ⊗ λ ∗ ◦ a n ◦ λ − ∗ = n X j =0 ( − j ρ ( λ )( a ⊗ · · · ⊗ a j a j +1 ⊗ · · · ⊗ a n )= ρ ( λ )( n X j =0 ( − j a ⊗ · · · ⊗ a j a j +1 ⊗ · · · ⊗ a n )= ρ ( λ ) d n ( a ⊗ · · · ⊗ a n ) verifies that the Hochschild differential d is U(1) -equivariant and therefore ultimately ρ extends to an actionby DG-algebra automorphisms on the chain complex ( b C • , d ) . Next, define a mapping ˆ P n : b C n −→ b C n by a ⊗ · · · ⊗ a n Z U(1) ρ ( λ )( a ⊗ · · · ⊗ a n ) dµ ( λ ) , where dµ ( λ ) is the Haar measure on the Lie group U(1) . Since the group is compact the integral is convergentand consequently the map ˆ P n is well-defined. The map is C -linear, its image Im( ˆ P n ) is the fixed pointsubspace b C U(1) n and its kernel ker( ˆ P n ) is simply the complement of b C U(1) n . It is a straightforward computationto show that ˆ P n − ◦ d n = d n − ◦ ˆ P n which implies that the mapping ˆ P : b C • −→ b C • is a chain complexendomorphism. By the axiom of choice one can decompose the complex vector space b C • into the subspaces(4) b C • = Im( ˆ P ) ⊕ ker( ˆ P ) = b C U(1) • ⊕ ker( ˆ P ) . The
U(1) -equivariance of the Hochschild differential d has as a consequence d ( b C U(1) n ) ⊆ b C U(1) n − and d (ker( ˆ P n )) ⊆ ker( ˆ P n − ) . This means that ( b C U(1) • , d ) and (ker( ˆ P ) , d ) are subcomplexes of ( b C • , d ) with (ker( ˆ P • ) , d ) havinga free U(1) -action.The rest of the proof cann be split in the following steps:
Step : We prove that the inclusion morphism i : b C U(1) • ֒ → b C • is in fact a quasi-isomorphism by showing TRACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS that the mapping cone chain complex cone( i ) · · · · · · b C U(1) n b C U(1) n − b C U(1) n − · · · · · · M M M · · · · · · b C n +1 b C n b C n − · · · · · · i n +1 d n +1 i n d n i n − d n − i n − d n − d n +1 d n +1 D n +1 d n D n d n − of the inclusion morphism i with cone( i ) n = b C U(1) n − ⊕ b C n and a differential D n = − d n − − i n − d n ! is acyclic.To demonstrate this it suffices to check that ker( D n ) ⊆ Im( D n +1 ) for every n ∈ Z ≥ . It is obvious that anarbitrary element ( v, w ) ∈ ker( D n ) = (cid:8) ( v, w ) ∈ b C U(1) n − ⊕ b C n | d n − ( v ) = 0 and d n ( w ) = i n − ( v ) (cid:9) satisfies the condition d n ( w ) ∈ b C U(1) n − . Writing w as a sum of an element w ∈ b C U(1) n and an element z ∈ ker( P n ) in line with (5) implies consequently ρ ( λ ) d n ( w ) = d n ( w ) + ρ ( λ )( d n ( z )) = d n ( w ) + d n ( z ) for every element λ ∈ U(1) and subsequently d n ( z ) = 0 . Thus, w can be expressed as a sum(5) w = w + z, where w ∈ b C U(1) n and z ∈ b Z n ∩ ker( P n ) .Taking the differential of ρ : U(1) −→ Aut(( b C • , d )) at the identity of U(1) yields the action ρ ∗ : u (1) −→ End(( b C • , d )) of the Lie algebra u (1) on the chain complex ( b C • , d ) . For any ǫ = iθ ∈ u (1) ∼ = i R the image ρ ∗ ( ǫ ) is a DG-algebra endomorphism of ( b C • , d ) given by(6) ρ ∗ ( ǫ )( a ⊗ · · · ⊗ a n ) := ddt (cid:12)(cid:12) t =0 ρ ( e ǫt )( a ⊗ · · · ⊗ a n ) = n X j =0 a ⊗ · · · ⊗ [ E, a j ] ⊗ · · · ⊗ a n , for any a ⊗ · · · ⊗ a n ∈ b C n , where E = ǫ P dim h k =1 y k ∂∂y k is the Euler field on h corresponding to ǫ ∈ u (1) . Wenotice that the free U(1) -action ρ induces a free u (1) -action ρ ∗ on ker( ˆ P • ) . In what follows, we show thatthe image ρ ∗ ( i ) of the sole generator of u (1) is a null-homotopic endomorphism of the chain complex b C • .For that purpose we define a C -linear map h : b C n −→ b C n +1 by(7) a ⊗ · · · ⊗ a n n X j =0 ( − j +1 a ⊗ · · · ⊗ a j ⊗ E ⊗ a j +1 ⊗ · · · ⊗ a n , where again E stands for the Euler vector field on h associated to the generator i . Let a ⊗ · · · ⊗ a n ∈ b C n .Then we compute dh ( a ⊗ · · · ⊗ a n ) = d (cid:0) P nj =0 ( − j +1 a ⊗ · · · ⊗ a j ⊗ E ⊗ a j +1 ⊗ · · · ⊗ a n (cid:1) = P nj =0 ( − j +1 (cid:8) ( − j a ⊗ · · · ⊗ a j E ⊗ a j +1 ⊗ · · · ⊗ a n +( − j +1 a ⊗ · · · ⊗ a j ⊗ Ea j +1 ⊗ · · · ⊗ a n + P j − k =0 ( − k a ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ ( a j ⊗ E ⊗ a j +1 ) ⊗ · · · ⊗ a n − P nk = j +1 ( − k a ⊗ · · · ⊗ ( a j ⊗ E ⊗ a j +1 ) ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a n (cid:9) = P nj =0 (cid:0) a ⊗ · · · ⊗ Ea j +1 ⊗ . . . a n − a ⊗ · · · ⊗ a j E ⊗ · · · ⊗ a n (cid:1) + P nj =0 P j − k =0 ( − j + k +1 a ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ ( a j ⊗ E ⊗ a j +1 ) ⊗ · · · ⊗ a n RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 9 − P nj =0 P nk = j +1 ( − j + k +1 a ⊗ · · · ⊗ ( a j ⊗ E ⊗ a j +1 ) ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a n . (8)as well as hd ( a ⊗ · · · ⊗ a n ) = h ( P nk =0 ( − k a ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a n )= P k − j =0 P nk =0 ( − j + k +1 a ⊗ · · · ⊗ a j ⊗ E ⊗ a j +1 ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a n − P nj = k +1 P nk =0 ( − j + k +1 a ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a j ⊗ E ⊗ a j +1 ⊗ · · · ⊗ a n = − P nj =0 P j − k =0 ( − j + k +1 a ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a j ⊗ E ⊗ a j +1 ⊗ · · · ⊗ a n + P nj =0 P nk = j +1 ( − j + k +1 a ⊗ · · · ⊗ a j ⊗ E ⊗ a j +1 ⊗ · · · ⊗ a k a k +1 ⊗ · · · ⊗ a n (9)The sum of (8) and (9) yields ( dh + hd )( a ⊗ · · · ⊗ a n )= P nj =0 ( a ⊗ · · · ⊗ Ea j +1 ⊗ . . . a n − a ⊗ · · · ⊗ a j E ⊗ · · · ⊗ a n )= P n − j =0 a ⊗ · · · ⊗ Ea j +1 ⊗ · · · ⊗ a n − P nj =1 a ⊗ · · · ⊗ a j E ⊗ · · · ⊗ a n + Ea ⊗ · · · ⊕ a n − a E ⊗ · · · ⊗ a n = P nj =1 a ⊗ · · · ⊗ Ea j ⊗ · · · ⊗ a n − P nj =1 a ⊗ · · · ⊗ a j E ⊗ · · · ⊗ a n +[ E, a ] ⊗ · · · ⊗ a n = P nj =0 a ⊗ · · · ⊗ [ E, a j ] ⊗ · · · ⊗ a n , (10)which by Definition (6) is equal to ρ ∗ ( i )( a ⊗ · · · ⊗ a n ) . Hence h is a contraction map and ρ ∗ ( i ) is null-homotopic map. Since ρ ∗ is a free u (1) -action on ker( ˆ P • ) its kernel is by definition trivial and consequently ρ ∗ ( i ) is an injective chain complex map which is invertible on Im ρ ∗ ( i ) ⊆ b C n . Take an n -cycle z ∈ ker( ˆ P n ) ∩ b Z n . Computation (10) entails that ρ ∗ ( i )( z ) = dh ( z ) . Since ρ ∗ ( i ) has an inverse on Im ρ ∗ ( i ) , we have z = ρ ∗ ( i ) − dh ( z ) = dρ ∗ ( i ) − ( hz ) ∈ ker( P n ) ∩ b B n . Consequently, (5) can be made more precise. Namely, if ( v, w ) ∈ ker( D n ) , then w = w + z where w ∈ b C U(1) n and z ∈ b B n ∩ ker( ˆ P n ) . There always are − x ∈ b C U(1) n and y ∈ b C n +1 such that w = d n +1 ( y ) − i n ( x ) . Since d n ( w ) = − d n ( i n ( x )) = − i n − ( d n ( x )) = − i n − ( v ) , then the injectivity of i n − stipulates that v = − d n ( x ) . This has as a consequence that every ( v, w ) ∈ ker( P n ) can be penned down as ( v, w ) = ( − d n ( x ) , d n +1 ( y ) − i n ( x )) for an appropriate x ∈ b C U(1) n and y ∈ b C n . This implies what is neededto ascertain that ˆ i • : b C U(1) • ֒ → b C • is a quasi-isomorphisms, namely ker( D n ) ⊆ Im( D n +1 ) = ( − d n ( x ) d n +1 ( y ) − i n ( x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) xy ! ∈ b C U(1) n ⊕ b C n +1 ) . Step 2 : In an analogous manner we demonstrate that the inclusion morphisms i • : C U(1) • ֒ → C • is a quasi-isomorphism. Step 3 : Lastly, we show that b C U(1) • = C U(1) • . The U(1) -action on b D ( h ) ⋊ G and 𝒟 ( h ) ⋊ G is determined by the action on their respective generators { y i ∈ C [ h ] , ∂∂y i , C G | i = 1 , . . . , dim h } : y i ρ ( λ ) λy i∂∂y i ρ ( λ ) λ − ∂∂y i g ρ ( λ ) g It is evident that the
U(1) -invariant subspaces of b 𝒟 ( h ) ⋊ G and 𝒟 alg. ( h ) ⋊ G are both generated by C G andthe subspace spanned by { y i ∂∂y j | i, j = 1 , . . . , dim h } and are therefore identical. Step 1, Step 2 in combination with
Step 3 are related by the diagram b C • i ←−−−− ֓ b C U(1) • = C U(1) • i ֒ −−−−→ C • . which at the level of homologies provides us with the statement of the proposition. (cid:3) The first direct corollary of that proposition is the next lemma.
Corollary 3.2.
Let 𝒟 ( U ) be the algebra of holomorphic differential operators on a connected G -invariant openset U in the analytic topology of a complex vector space h . Then, HH j ( 𝒟 ( U ) ⋊ G ) ∼ = HH j ( 𝒟 alg. ( h ) ⋊ G ) . Proof.
Consider the embeddings of chain complexes i : C • ( 𝒟 alg. ( h ) ⋊ G ) ֒ → C • ( 𝒟 ( U ) ⋊ G ) and i : C • ( 𝒟 ( U ) ⋊ G ) ֒ → b C • ( b D ( h ) ⋊ G ) . The composition I := i ◦ i fits in the following commutative diagram b C • ( 𝒟 alg. ( h ) ⋊ G ) U(1) b C • ( b 𝒟 ( h ) ⋊ G ) U(1) C • ( 𝒟 alg. ( h ) ⋊ G ) b C • ( b 𝒟 ( h ) ⋊ G ) = quasi-iso. quasi-iso. I where the upper horizontal map is the identity from Step 3 and the vertical maps are the injective quasi-isomorphisms from the proof of Proposition 3.1. At the level of Hochschild homology, I ∗ is an isomorphismwhence i ∗ is surjective. Since i ∗ is clearly injective, too, the claim follows. (cid:3) Remark . The above result remains valid for disjoint unions of connected G -invariant open sets U i in G -spaces h i . Namely, it can be shown that HH j ( 𝒟 ( ` i U i ) ⋊ G ) ∼ = HH j ( 𝒟 alg. ( ` i h i ) ⋊ G ) .From now on for notational brevity let us designate the field of Laurent series C (( ~ )) by K . The formal pa-rameter ~ is a central element in the formal Cherednik algebra b H , ~ ( h , G ) . Hence, one can localize the under-lying non-commutative ring with respect to ~ the same way as any other commutative ring. Let b H , (( ~ )) ( h , G ) denote the localization of the degree-wise formal completion b H , ~ ( h , G ) with respect to the multiplicativesubmonoid S := { ~ n | n ∈ Z > } , that is, b H , (( ~ )) ( h , G ) = b H , ~ ( h , G )[ S − ] = b H , ~ ( h , G ) ⊗ C [[ ~ ]] K . Sim-ilarly, 𝒟 alg. ( h ) ⋊ G (( ~ )) is isomorphic to the localization of Weyl ~ ( h ⊕ h ∗ ) ⋊ G with respect to S where Weyl ~ ( h ⊕ h ∗ ) denotes the homogenization of the Weyl algebra Weyl( h ⊕ h ∗ ) by means of a formal param-eter ~ with deg( ~ ) = 1 . Let C [ ε ] with ε = 0 be the ring of dual numbers. The following proposition is aconsequence of Theorem 2.6 and Proposition 3 and the proof of its first half mimics and expounds the proofof Proposition 1 in [RT12]. RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 11
Proposition 3.4. i) HH j (cid:0) b H , (( ~ )) ( h , G ) (cid:1) ∼ = HH j (cid:0) 𝒟 alg. ( h ) ⋊ G (( ~ )) (cid:1) ∼ = K a ( G ) j , where a ( G ) j denotes the number of conjugacy classes of elements of G having eigenvalue with multiplicity j . ii) HC j (cid:0) b H , (( ~ )) ( h , G ) ⊗ C C [ ε ] (cid:1) = HC j (cid:0) 𝒟 alg. ( h ) ⋊ G (( ~ )) ⊗ C C [ ε ] (cid:1) Proof. i)
The C [[ ~ ]] -algebra K of formal Laurent series has an increasing and exhaustive ring filtration by C [[ ~ ]] -modules C [[ ~ ]] ⊂ C [[ ~ ]] ~ − ⊂ · · · ⊂ C [[ ~ ]] ~ − p ⊂ · · · ⊂ K , which gives rise to a directed system ( C [[ ~ ]] ~ − p , τ pq ) in the category of C [[ ~ ]] -modules where τ pq : C [[ ~ ]] ~ − p ֒ → C [[ ~ ]] ~ − q is the natural em-bedding for p, q ∈ Z ≥ with p ≤ q . For each pair p, q ∈ Z ≥ so that p ≤ q the natural inclusion map I p : C [[ ~ ]] ~ − p ֒ → K satisfies I p = I q ◦ τ pq and the assignment P i = − p a i ~ i P i = − p a i ~ i defines amorphisms θ : K → lim −→ p C [[ ~ ]] ~ − p of C [[ ~ ]] -modules such that θ ◦ I p = i p for every p ∈ Z ≥ , where i p : C [[ ~ ]] ~ − p ֒ → lim −→ p C [[ ~ ]] ~ − p is the natural inclusion map into the injective limit of the directed system.Consequently, K satisfies the same universal property as the direct limit. Thus, by uniqueness of the universalproperty of lim −→ p C [[ ~ ]] ~ − p there is an isomorphism of C [[ ~ ]] -modules(11) K ∼ = lim −→ p C [[ ~ ]] ~ − p . Let b K n := b K n ( b H , ~ ( h , G )) designate the n -th Hochschild chain complex C [[ ~ ]] -module of b H , ~ ( h , G ) . Thentaking account of isomorphism (11), we recast the n -th Hochschild chain complex K -module of b H , (( ~ )) ( h , G ) in the following form: b C n ( b H , (( ~ )) ( h , G )) ∼ = lim −→ p b K n ( b H , ~ ( h , G )) ~ − p . (12)The Hochschild homology of b H , (( ~ )) ( h , G ) is obtained by applying the homology functor H • on the chaincomplex b C • ( b H , (( ~ )) ( h , G )) : HH • ( b H , (( ~ )) ( h , G )) = lim −→ p HH • ( b H , ~ ( h , G )) ~ − p . (13)It remains to compute HH • ( b H , ~ ( h , G ) , b H , ~ ( h , G )) . For this aim, we define an increasing, exhaustive andbounded above C [[ ~ ]] -module filtration on b K • := b K • ( b H , ~ ( h , G )) by F p b K • = b K • ~ − p for p ∈ Z ≤ . Theobservation b K n = b H , ~ ( h , G ) ˆ ⊗ C [[ ~ ]] . . . ˆ ⊗ C [[ ~ ]] b H , ~ ( h , G )= C [[ h ]] ⊗ C [ h ] T • ( h ⊕ h ∗ ) ⊗ C C G [ ~ ] ⊗ C [ ~ ] C [[ ~ ]] J ! ˆ ⊗ C [[ ~ ]] n + 1= C [[ h ]] ⊗ C [ h ] T • ( h ⊕ h ∗ ) ⊗ C C G [ ~ ] J ! ˆ ⊗ C n + 1 ⊗ C [ ~ ] C [[ ~ ]] combined with the fact that C [[ ~ ]] / ~ p C [[ ~ ]] ∼ = C [ ~ ] / ~ p C [ ~ ] demonstrates that the chain complex is completein the ~ -adic topology and hence that the filtration on b K • is complete. The filtration of the chain complex b K • determines by the construction theorem a spectral sequence { E rpq } starting with E pq = F p b K p + q /F p − b K p + q ∼ = ~ − p b H , ~ ( h , G ) ~ b H , ~ ( h , G ) ˆ ⊗ C . . . ˆ ⊗ C b H , ~ ( h , G ) ~ b H , ~ ( h , G ) ∼ = ~ − p b 𝒟 ( h ) ⋊ G ˆ ⊗ C . . . ˆ ⊗ C b 𝒟 ( h ) ⋊ G = ~ − p b C p + q ( b 𝒟 ( h ) ⋊ G ) for p ∈ Z ≤ for p ∈ Z > , (14)and consequently E pq = H p + q ( E p ∗ )= ~ − p HH p + q ( b 𝒟 ( h ) ⋊ G )= ~ − p C a ( G ) p + q for p ∈ Z ≤ for p ∈ Z > , (15)where in the second line we apply Proposition 3.1. Since for each degree n = p + q , it follows from (14)that E pq = 0 for every p > , the spectral sequence { E rpq } is bounded from above. In the generic situation h is a completely reducible G -module which can be decomposed in a direct sum h = h ⊕ m ⊕ . . . h ⊕ m k k ofirreducible G -submodules h j , j = 1 , . . . , k of h , with multiplicity m j and one can correspondingly express 𝒟 alg. ( h ) ⋊ G as 𝒟 alg. ( h ) ⋊ G ∼ = (cid:0) 𝒟 alg. ( h ) ⋊ G (cid:1) ⊗ C m ⊗ C · · · ⊗ C (cid:0) 𝒟 alg. ( h k ) ⋊ G (cid:1) ⊗ C m k . Finally,
Künneth’s formula for chain complexes yields the isomorphism HH p + q ( 𝒟 alg. ( h ) ⋊ G ) ∼ = M k P j =0 mj P sj =1 α jsj = p + q ( O j =0 ,...,ks j =1 ,...,m j HH α jsj ( 𝒟 alg. ( h j ) ⋊ G ) ) . (16)According to formula (2.12) in [EG02] for every irreducible G -submodule h j , we have that a ( G ) α jsj = 0 when α js j is odd. Consequently, HH α jsj ( 𝒟 alg. ( h j ) ⋊ G ) = 0 when α js j is odd. We notice that whenever p + q is odd, in every summand of the right hand side of (16) at least one index α js j is odd. This entails that HH p + q ( 𝒟 alg. ( h ) ⋊ G ) = 0 for p + q odd number. Thus, E pq = E ∞ pq , which renders the sequence regular. The complete convergence theorem implies that E rpq converges to HH p + q ( b H , ~ ( h , G )) , that is, E pq ∼ = F p HH p + q ( b H , ~ ( h , G )) F p − HH p + q ( b H , ~ ( h , G )) ∼ = C [[ ~ ]]( ~ ) ⊗ C [[ ~ ]] ~ − p HH p + q ( b H , ~ ( h , G ))= ~ − p HH p + q ( b H , ~ ( h , G )) . (17)Equating (15) with (17) yields HH p + q ( b H , ~ ( h , G )) = C [[ ~ ]] a ( G ) p + q . Claim i) of the proposition follows im-mediately from (13). ii) Since the ring of dual numbers is a flat C -module, the completed Dunkl embedding induces an injec-tive map g : b H , (( ~ )) ( h , G ) ⊗ C [ ε ] ֒ → b 𝒟 ( h ) ⋊ G (( ~ )) ⊗ C [ ε ] . Consequently, it induces an injective map ofHochschild chain complexes b C • (cid:0) b H , (( ~ )) ( h , G ) ⊗ C [ ε ] (cid:1) ֒ → b C • (cid:0) b 𝒟 ( h ) ⋊ G (( ~ )) ⊗ C [ ε ] (cid:1) which in turn inducesan embedding between the respective Hochschild homologies. Künneth’s formula, Proposition 3 and claim i) imply that the Hochschild homologies of b H , (( ~ )) ( h , G ) ⊗ C [ ε ] and b 𝒟 ( h ) ⋊ G (( ~ )) ⊗ C [ ε ] are isomorphic. RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 13
Hence, the map g induces an isomorphism in Hochschild homology. Consequently, according to Corollary2.2.3 in [Lod13] the algebra map g induces an isomorphism in cyclic homology. (cid:3) The space of traces on b H , (( ~ )) ( h , G ) is isomorphic to the zeroth cyclic cohomology group of b H , (( ~ )) ( h , G ) which in tern is isomorphic to the dual of the zeroth Hochschild homology K -module HH ( b H , (( ~ )) ( h , G )) . Asthe Hochschild homology of b H , (( ~ )) ( h , G ) is finite-dimensional, the space of K -linear traces on b H , (( ~ )) ( h , G ) coincides with the zeroth Hochschild homology K -module HH ( b H , (( ~ )) ( h , G )) . The ensuing corollary showsthat at least under some circumstances there are non-trivial traces on b H , (( ~ )) ( h , G ) . Corollary 3.5.
Suppose G ⊂ GL( h ) is a well-generated complex reflection group with h G = { } . Then, thedimension of d HH ( b H , (( ~ )) ( h , G )) over K is at least .Proof. Theorem 2.1 along with condition h G = { } and Lemma 2.2 imply that supp( G ) = h ⊕ · · · ⊕ h m = span S { α ∨ s } , where h i is an irreducible G i -module for every i = 1 , . . . , m and α ∨ s is the root of thecomplex reflection s . Since G is well-generated, that is, rank( G ) = | S | , we have that h ⊕ · · · ⊕ h m = L mi =1 L s i ∈ S i span { α ∨ s i } . Since from Theorem 2.1 we know that each irreducible complex reflection sub-groups G i is generated by those complex reflections in G whose roots belong to h i , it follows that each G i for , . . . , m , is an irreducible, well-generated complex reflection group. Consequently, by Lemma 2.4 each G i possesses a Coxeter element c i . Take c := ( c , . . . , c m ) . This group element in G × · · · × G m has noeigenvalue equal to and corresponds to an element in G with no eigenvalue equal to one. Thus, a ( G ) ≥ in this case. The claim follows then by Theorem 3.4. (cid:3) Let 𝒜 H, (( ~ )) n − l,l denote from now on the K -algebra b D (( ~ )) n − l ˆ ⊗ K b H , (( ~ )) ( C l , H ) where b D (( ~ )) n − l is the algebra ofdifferential operators on the formal neighborhood of zero in K n . With the help of the previous results weeasily arrive at the following result about the Hochschild homology of 𝒜 H, (( ~ )) n − l,l and the cyclic homology ofthe Z -graded algebra 𝒜 H, (( ~ )) n − l,l ⊗ C C [ ε ] . Corollary 3.6.
For every m ∈ Z ≥ , i ) HH m (cid:0) 𝒜 H, (( ~ )) n − l,l (cid:1) ∼ = HH m − n +2 l (cid:0) 𝒟 alg. ( C l ) ⋊ H (( ~ )) (cid:1) ∼ = K a ( H ) m − n +2 l where a ( H ) m − n +2 l is as in Proposition 3.4, i) . ii ) HC m (cid:0) 𝒜 H, (( ~ )) n − l,l ⊗ C C [ ε ] (cid:1) ∼ = HC m − (2 n − l ) (cid:0) 𝒟 alg. ( C l ) ⋊ H (( ~ )) ⊗ C C [ ε ] (cid:1) ∼ = M γ ∈ Conj( H ) (cid:0) HC m − (2 n − l ) − k γ ( K ) ⊕ K · [ ε ⊗ · · · ⊗ ε | {z } m − (2 n − l ) − k γ − times ] (cid:1) where k γ := dim( C l ⊕ C l ∗ ) γ , · denotes multiplication and [ ε ⊗ · · · ⊗ ε ] is the cohomology class of ε ⊗ · · · ⊗ ε .Proof. i) This follows from Künneth’s formula, Proposition 3.1, Theorem 2 in [Wod87] and finally Proposition3.4, i) . ii) Without any loss of generality assume in the following for the sake of simplicity that deg( ε ) = 0 .Since the ring of dual numbers is a flat C -module, the tensor product over C with C [ ε ] preserves the nat-ural injection b H , (( ~ )) ( C l , H ) ֒ → 𝒜 H, (( ~ )) n − l,l and we obtain an injective embedding of Z -graded algebras f : b H , (( ~ )) ( C l , H ) ⊗ C [ ε ] ֒ → 𝒜 H, (( ~ )) n − l,l ⊗ C [ ε ] . In turn it induces an injection of Hochschild chain complexes f : b C • (cid:0) b H , (( ~ )) ( C l , H ) ⊗ C [ ε ] (cid:1) ֒ → b C • +(2 n − l ) (cid:0) 𝒜 H, (( ~ )) n − l,l ⊗ C [ ε ] (cid:1) by (cid:0) a ⊗ x + y ε (cid:1) ⊗ · · · ⊗ (cid:0) a m ⊗ x m + y m ε (cid:1) ⊗ (2 n − l ) ⊗ (cid:0) a ⊗ x + y ε (cid:1) ⊗ · · · ⊗ (cid:0) a m ⊗ x m + y m ε (cid:1) which by abuse of notation we keep calling f . The corresponding induced map f ∗ : HH • (cid:0) b H , (( ~ )) ( C l , H ) ⊗ C [ ε ] (cid:1) → HH • +(2 n − l ) (cid:0) 𝒜 H, (( ~ )) n − l,l ⊗ C [ ε ] (cid:1) is obviously injective, too. Indeed, assume f ∗ (cid:16)(cid:0) a ⊗ x + y ε (cid:1) ⊗ · · ·⊗ (cid:0) a m ⊗ x m + y m ε (cid:1)(cid:17) is an m + (2 n − l ) -boundary. This implies that there are c , . . . , c m +(2 n − l )+1 ∈ 𝒜 H, (( ~ )) n − l,l ⊗ C [ ε ] such that f ∗ (cid:16)(cid:0) a ⊗ x + y ε (cid:1) ⊗ · · · ⊗ (cid:0) a m ⊗ x m + y m ε (cid:1)(cid:17) = 1 ⊗ · · · ⊗ ⊗ m +(2 n − l )+1 X r =2 n − l ( − r c n − l +1 ⊗ · · · ⊗ c r c r +1 ⊗ · · · ⊗ c m +(2 n − l )+1 = 1 ⊗ · · · ⊗ ⊗ m +1 X r =0 ( − r c ⊗ · · · ⊗ c r c r +1 ⊗ · · · ⊗ c m +1 . From that we deduce that (cid:0) a ⊗ x + y ε (cid:1) ⊗· · ·⊗ (cid:0) a m ⊗ x m + y m ε (cid:1) is an m -boundary. By Künneth’s formula andclaim i) the induced map is an isomorphism. Then, by virtue of Corollary 2.2.3 in [Lod13] the map f inducesan isomorphism in cyclic homology which combined with Proposition 3.4, ii) yields the first isomorphism inclaim ii) of this proposition. The second one is implied by Isomorphism (A.13) HC • (Weyl ~ ( C l ⊕ C l ∗ )[ S − ] ⋊ H ⊗ C [ ε ]) ∼ = ⊕ γ ∈ Conj( H ) HC •− k γ ( K [ ε ]) in [PPT07] accounting that HC • ( K [ ε ]) ∼ = (cid:0) HC • ( K ) ⊕ K · [ ε ⊗• ] (cid:1) in (A.4) in [PPT07]. (cid:3)
4. Trace densities and hypercohomology
In this section we introduce trace density maps for the sheaf of Cherednik algebras from [Eti04] makinguse of the construction of ℋ ,c,X,G in the proof of [Vit19, Theorem 6.3]. Subsequently by means of thesemaps we construct a morphism from the hypercohomology of the sheaf of Hochschild chain complexes of ℋ ,c,X,G to the the Chen-Ruan orbifold cohomology of the global quotient X/G . The trace density mapsgo through in the case when the complex valued parameters c are replaced by a formal parameter ~ and weshow that the map between the hypercohomology of the Hochschild chain complexes of ℋ , (( ~ )) ,X,G and theChen-Ruan orbifold cohomology with coefficients in K becomes an isomorphsim.4.1. Hochschild n − l -cocycle of the Harish-Chandra modules 𝒜 Hn − l,l and 𝒜 H, (( ~ )) n − l,l . Assume that H is such that HH ( H ,c ( C l , H )) is non-trivial. Consequently, since HH • ( H ,c ( C l , H )) is embedded in d HH • ( b H ,c ( C l , H )) , the complex dimension of the trace group of b H ,c ( C l , H ) is at least . This implies thatthere are non-trivial linear functionals on b H ,c ( C l , H ) . Remark . For instance, when H = S n , ( n ≥ and for generic values of c the trace group of H ,c ( C l , H ) is non-trivial ( cf. e.g. [BEG04]).In the formal case by Corollary 3.5 there are non-trivial linear functionals on b H , (( ~ )) ( C l , H ) if H is awell-generated finite complex reflection group with no fixed points on C l , for example. RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 15
Let φ , respectively , φ ~ be non-trivial linear functionals on b H ,c ( C l , H ) and b H , (( ~ )) ( C l , H ) , respectively. Wedefine a Hochschild n − l -cocycle ψ n − l of 𝒜 Hn − l,l by(18) ψ n − l ( a ⊗ b ⊗ · · · ⊗ a n − l ⊗ b n − l ) := τ n − l ( a ⊗ · · · ⊗ a n − l ) φ ( b . . . b n − l ) for a , . . . , a n − l ∈ b D n − l , b , . . . , b n − l ∈ b H ,c ( C l , H ) where τ n − l is the version of the gl n − l ( C ) -basic reduced Hochschild n − l -cocycle of b D n − l emplyed in [EF08]. With the same notation we definea Hochschild n − l -cocycle ψ ~ n − l of 𝒜 H, (( ~ )) n − l,l by(19) ψ ~ n − l ( a ⊗ b ⊗ · · · ⊗ a n − l ⊗ b n − l ) := τ ~ n − l ( a ⊗ · · · ⊗ a n − l ) φ ~ ( b . . . b n − l ) where τ ~ n − l is the gl n − l ( K ) -basic reduced Hochschild n − l -cocycle of b D (( ~ )) n − l constructed in [FFS05]. Thefollowing proposition is analogous to Proposition in [RT12]. Proposition 4.2.
The Hochschild n − l -cocycles ψ n − l and ψ ~ n − l are gl n − l ( C ) ⊕ z -basic and ( gl n − l ( C ) ⊕ z ) ⊗ K -basic, respectively.Proof. The proof is by verification and is the same as the proof of Proposition in [RT12]. (cid:3) Construction of the trace density maps.
Fix a stratum X iH in X with codim( X iH ) = l correspondingto a subgroup H of G and let X Hi be the connected component of the fixed point submanifold of H in X containing X iH . Let the maps j iH : X iH ֒ → X and j Hi : X iH ֒ → X be the respective canonical inclusions. Let 𝒩 coor be the bundle of infinite jets of parametrizations of the normal bundle N to the fixed point submanifold X Hi . In [Vit19, Section 4, Section 5] it was shown that there is a flat holomorphic connection -form ω on 𝒩 coor over X iH with values in 𝒜 Hn − l,l . It can trivially be defined over the whole fixed point submanifold X Hi .For every ind GH W x from ℬ GX , we have by [Vit19, Theorem 6.3] H ,c (ind GH W x ) −→ π coor ∗ 𝒪 flat ( 𝒩 coor | X iH × 𝒜 Hn − l,l )( W ix,H ) (20) ( q, X g,g ′ ∈ G/H g ⊗ t gg ′ ⊗ t ′ ) id G ⊗ t HH ⊗ id G . For a ind GK W x in ℬ GX with x lying on a stratum X jK , which is contained in with X Hi , the assignment(21) H ,c (ind GK W x ) → π coor ∗ 𝒪 flat ( 𝒩 coor | X iH × 𝒜 Hn − l,l )( W ix,H ) is defined in a more subtle fashion. Note that by Cartan’s Lemma apart from X K the slice W x intersects onlyisotropy types X L associated to subgroups L of K . Because of this and by [Vit19, Corollary B.9], the map(21) is the composition of the following maps H ,c (ind GK W x , G ) ∼ = −→ C G ⊗ C K H ,c ( W x , K ) ⊗ C K C G → H ,c ( W x \ X jK , K ) ∼ = −→ lim ←− y ∈ X L ∩ W x H ,c (ind KL W y , K ) ։ Y L The morphism (23) has a unique extension (24) ¯ p : ℋ ,c,X,G → j Hi ∗ π coor ∗ 𝒪 flat ( 𝒩 coor × 𝒜 Hn − l,l ) . Proof. We distinguish two cases: ) the codimension of the stratum X iH is equal to or bigger than , ) thestratum X iH is the prinicipal (dense and open) stratum in X . ) Let W x be K -invariant linear slice centered on a stratum X jK contained in X Hi as above. By Hartog’sTheorem the third arrow in (22) is surjective. Hence, the images of the morphisms (22) and H , ˜ c ( W x \ X jK , H ) ∼ = −→ lim ←− W y ,y ∈ X L ∩ W x ,L Let G ′ be a Coxeter group and G a complex reflection group acting on the complex manifold X .Then, i) there is a non-trivial morphism H −• ( X/G ′ , b C • ( ℋ ,c,X,G ′ )) → H n −• CR ( X/G ′ , C ) ii) there is an isomorphism H −• ( X/G, b C • ( ℋ , (( ~ )) ,X,G )) → H n −• CR ( X/G, C (( ~ ))) .Proof. i) Consider the natural map(28) M ig ∈ G ′ χ gi : b C • ( ℋ ,c,X,G ′ ) −→ (cid:0) M ig ∈ G ′ j ∗ Ω n − X gi ) −• X gi (cid:1) G ′ induced by the trace density maps (26). Each χ gi is non-zero since each cyclic subgroup h g i is by definitionisomorphic to S ( cf. Remark 4.1). Taking the hypercohomology functor H −• on both sides of (28) andaccounting that the smooth de Rham complex is a complex of soft and hence Γ -acyclic sheaves, we get a map M ig ∈ G ′ χ iH ∗ : H −• ( X/G ′ , b C • ( ℋ ,c,X,G ′ )) → (cid:0) M ig ∈ G ′ H n − X gi ) −• dR ( X gi , C ) (cid:1) G ′ where the cohomologies on the right hand side are de Rham cohomologies. Since the right hand side is iso-morphic to the Chen-Ruan cohomology of the global quotient orbifold X/G ′ with values in C , the assertionfollows. ii) Each group h g i is by definition isomorphic to a cyclic group which do not have non-trivial fixed pointelements in C l . Hence, the maps χ gi, (( ~ )) are non-zero ( cf. th paragraph in Subsection 4.1). As per definitionof the basis ℬ GX , for every x ∈ X iH , there is a contractible H -invariant slice W x in X , a H -invariant con-tractible set V in the product topology of C n , n = dim X , containing the origin of C n , and an H -equivariantbiholomorphism f : W x → V with f ( x ) = 0 . The differential of f equips C n with the structure of an H -representation. Similarly, each C n [ g ] := [ g, C n ] ⊂ G × H C n becomes a gHg − -space. As a result, theinduction set ind GH C n = G × H C n = ` g ∈ G/H C n [ g ] acquires a natural right H -action. Moreover, each trans-late gW x possesses a gHg − -equivariant biholomorphism f [ g ] from gW x to an open set V [ g ] in C n [ g ] , given by f [ g ] ( y ) = [ g, f ( g − y )] for every y ∈ gW x . Consequently, there is a G -equivariant biholomorphism F from ind GH W x to ind GH V given by F ( y ) = f [ g ] ( y ) for every y ∈ gW x and every g ∈ G/H .Set l ig = codim( X gi ) . Consider the formal mapping(29) M ig ∈ G χ gi, (( ~ )) : b C • ( ℋ , (( ~ )) ,X,G ) → (cid:0) M ig ∈ G j gi ∗ Ω n − l ig −• X gi (( ~ )) (cid:1) G To prove that this map is a quasi-isomorphism it suffices to show that the homology sheaves on both sides of(29) are isomorphic as presheaves in the basis of the G -equivariant topology. By an identical argumentationas in Proposition 3.4, i) we have for the homology presheaf on every ind GH W x ∈ ℬ GX on the left hand side ofEquation (29) that(30) HH • ( H , (( ~ )) (ind GH W x , G )) ∼ = HH • ( 𝒟 (ind GH W x ) ⋊ G )(( ~ )) . Then, the G -equivariant biholomorphism F and Remark 3.3 imply that(31) HH • ( 𝒟 (ind GH W x ) ⋊ G )(( ~ )) ∼ = HH p ( 𝒟 (ind GH V ) ⋊ G )(( ~ )) ∼ = HH • ( 𝒟 alg. (ind GH C n ) ⋊ G )(( ~ )) . On the other hand, denote by Z H ( h ) and C H ( h ) the centralizer of an element h in H , respectively its con-jugacy class in H and by H dR the algebraic de Rham cohomology. Then, by virtue of Frobenius’ theorem wecan further simplify HH • ( 𝒟 alg. (ind GH C n ) ⋊ G ) ∼ = (cid:0) ⊕ ig ∈ G H n − GH C n ) gi −• ((ind GH C n ) gi , C ) (cid:1) G RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 19 ∼ = (cid:0) ⊕ g ∈ G/H ⊕ k ∈ gHg − H n − C n [ g ] ) k −• (( C n [ g ] ) k , C ) (cid:1) G ∼ = (cid:0) C G ⊗ C H ⊕ k ∈ H H n − C n ) k −• dR (( C n ) k , C ) (cid:1) G ∼ = (cid:0) ⊕ k ∈ H H n − l k −• dR (( C n ) k , C ) (cid:1) H ∼ = ⊕ C H ( h ) ∈ Conj( H ) (cid:0) ⊕ k ∈ C H ( h ) H n − l k −• dR (( C n ) k , C ) (cid:1) H ∼ = ⊕ C H ( h ) ∈ Conj( H ) (cid:0) C H ⊗ C Z H ( h ) H n − l h −• dR (( C n ) h , C ) (cid:1) H ∼ = ⊕ C H ( h ) ∈ Conj( H ) (cid:0) H n − l h −• dR (( C n ) h , C ) (cid:1) Z H ( h ) ∼ = ⊕ C H ( h ) ∈ Conj( H ) (cid:0) H • ( 𝒟 alg. ( C n ) , 𝒟 alg. ( C n ) h ) (cid:1) Z H ( h ) (32)where i in the first line denotes the connected components, the first isomorphism follows directly fromPropositions 3 and 4 in [DE05]. From [FT10] we know that the homology H • ( 𝒟 alg. ( C n ) , 𝒟 alg. ( C n ) h ) is one-dimensional, spanned by the Hochschild n − l h -cycle c n − l h = X σ ∈ S n − lh ⊗ u σ (1) ⊗ · · · ⊗ u σ (2 n − l h ) where u i − = ∂ x i − , u i = x i . Ergo, from the isomorphisms (30), (31) and (32) we conclude that HH • ( H , (( ~ )) (ind GH W x , G ) is spanned by the vector ( c n − l h ) C H ( h ) ∈ Conj( H ) . On the other hand, there isa natural isomorphism HH ( H , (( ~ )) ( C l , h h i ) , H , (( ~ )) ( C l , h h i ) ∗ ) = HH ( 𝒟 alg. ( C l ) ⋊ h h i , 𝒟 alg. ( C l ) ⋊ h h i ∗ )(( ~ ))= (cid:0) ⊕ ord( h ) k =1 HH ( 𝒟 alg. ( C l ) , 𝒟 alg. ( C l ) h k ∗ ) (cid:1) h h i where ord( h ) is the order of h in G . Each group HH ( 𝒟 alg. ( C l ) , 𝒟 alg. ( C l ) h k ∗ ) is spanned by a h k -twistedtrace tr h k ( · ) , defined in [Fed00]. Hence, for each trace φ of H , (( ~ )) ( C l , h h i ) , we can make the identification(33) φ = ord( h ) X k =1 λ k tr h k ( · ) . Evaluation of the right hand side of (29) gives (cid:0) M ig ∈ G j ∗ Ω n − l ig −• X gi (( ~ ))(ind GH W x ) (cid:1) G = (cid:0) M C G ( h ) ∩ H = ∅ M g ∈ C G ( h ) j ∗ Ω n − l ig −• X gi (( ~ ))(ind GH W x ) (cid:1) G = M C G ( h ) ∩ H = ∅ (cid:0) j ∗ Ω n − l ih −• X hi (( ~ ))(ind GH W x ) (cid:1) Z G ( h ) In the second line we employed the fact that by Cartan’s Lemma W x intersects only one connected componentfrom each X g . Consequently, the cohomology of the right hand side of (29) is M C G ( h ) ∩ H = ∅ (cid:0) H n − l ih −• ( W ix, 5. Algebraic index theorem At last we prove an algebraic index theorem for the formal trace densities (27) following established tech-niques from previous works such as [FFS05], [EF08] and [PPT07] and [RT12].Throughout the section we adhere to the structure of and the notation in sections 2.4 and 4.3 in [RT12].Fix a stratum X iH and let in the following codim( X iH ) = l . Let us fix a number N >> n . Let further g := gl N ( 𝒜 H, (( ~ )) n − l,l ) and let h := gl n − l ( K ) ⊕ (cid:0) z ⊕ gl N ( C ) (cid:1) ⊗ K . By Section 3.1 and 3.2 in [FFS05] theHochschild cocycle ψ ~ n − l , defined by Equation (19), corresponds to a unique n − l -Lie cocycle Ψ n − l ∈ C n − l ( g , h ; g ∗ ) . Let the mapping ev : C n − l ( g , h ; g ∗ ) → C n − l ( g , h ; K ) be the evaluation at the identity.In order to formulate an index theorem we first have to compute [ev Ψ n − l ] .Let W (( ~ )) n − l be the Lie algebra Der( K [[ x , . . . , x n − l ]]) and let n = ( z ⊕ gl N ( C )) ⊗ K . Then, the combination ofthe obvious Lie algebra embedding h ֒ → W (( ~ )) n − l ⋊ ( z ⊕ gl N ( C ) ⊗ K ) ⊗ 𝒪 n − l with the Lie algebra embedding W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l ֒ → g , given by v + ( A, B ) ⊗ p ⊗ Φ (( h )) ( v + A ⊗ p ) + B ⊗ Φ (( ~ )) ( p ) , where Φ (( ~ )) is defined as in [Vit19, Proposition 4.4], allows us to view h as a Lie subalgebra of g . Selecta decomposition of g into a direct sum of h -modules g ∼ = h ⊕ g / h . This yields a projection of h -modules pr : g → h along g / h which can be interpreted as an h -equivariant projection. The amount by which thisprojection fails to be a Lie algebra homomorphism is measured by the curvature C ∈ Hom( V g , h ) definedin [FFS05] by C ( v, w ) := [pr( v ) , pr( w )] − pr([ v, w ]) for all v, w ∈ g . By means of C one can define the Chern-Weil homomorphism χ : S • ( h ∗ ) h → H • ( g , h ; K ) by χ ( P )( v ∧ · · · ∧ v k ) = 1 k ! X σ ∈ S k σ (2 i − <σ (2 i ) ( − σ P (cid:0) C ( v σ (1) , v σ (2) ) , . . . , C ( v σ (2 k − , v σ (2 k ) ) (cid:1) for every P ∈ S • ( h ∗ ) h . Now we prove several supporting propositions for the Chern-Weil homomorphismswhich are needed for the computation of the cohomology class of [ev Ψ n − l ] . We conclude the section withthe promised algebraic index theorem for ℋ , (( ~ )) ,X,G . RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 21 Proposition 5.1. The Chern-Weil homomorphism 𝒳 : S q ( h ∗ ) h −→ H q ( g , h ; K ) is an isomorphism for N >>n and q ≤ n − l + k where k = min γ ∈ Conj( H ) k γ and k γ is as in Corollary 3.6, ii) .Proof. Assume in what follows that deg( ε ) = 1 . By Theorem 10.2.5 in [Lod13] we have that(34) H m ( gl N ( 𝒜 H, (( ~ )) n − l,l ⊗ C [ ε ])) ∼ = (cid:0) 𝒮 • (HC • ( 𝒜 H, (( ~ )) n − l,l ⊗ C [ ε ])[1]) (cid:1) m for every m ≥ where 𝒮 • is the graded symmetric product defined for instance in A.1 in [Lod13] and ( · ) m denotes the m -th degree of a graded module. As explained in Appendix A in [PPT07], the left hand side of(34) can be written as(35) H m ( gl N ( 𝒜 H, (( ~ )) n − l,l ⊗ C [ ε ])) ∼ = ⊕ mp =0 H p ( g , S m − p g ε ) On account of Corollary 3.6, ii) , the right hand side of the isomorphism can be written as(36) M d ≥ M j + ··· + j d = m (cid:16) ⊕ γ HC j − (2 n − l +2 k γ ) − ( K [ ε ]) (cid:17) ⊗ · · · ⊗ (cid:16) ⊕ γ HC j d − (2 n − l +2 k γ ) − ( K [ ε ]) (cid:17) The isomorphism (34) is graded of degree . Hence, in particular it respects the grading in ε . Hence, it mapscohomology classes of degree m − p in ε to elements of degree m − p in ε . Hence, inserting (35) and (36) in(34) and comparing degrees of ε , we get for every p ≤ m , H p ( g , S m − p g ε ) ∼ = (cid:0) ⊕ γ HC m − (2 n − l +2 k γ ) − ( K [ ε ]) (cid:1) m − p − th degree in ε ∼ = (cid:16) ⊕ γ (cid:0) HC m − (2 n − l +2 k γ ) − ( K ) ⊕ K [ ε ⊗ · · · ⊗ ε | {z } m − (2 n − l + 2 k γ ) -times ] (cid:1)(cid:17) m − p − th degree in ε ∼ = L γ ∈ Conj( H ) k γ = k K [ ε ⊗ · · · ⊗ ε | {z } m − n − l + 2 k ] , if p = 2 n − l + 2 k , if p < n − l + 2 k Since Lie algebra homology and cohomology are dual, another way of stating the above is: H p ( g , S q g ) isisomorphic to K a ( H ) n − l + k when p = 2 n − l + 2 k and is otherwise. The remainder of the proof followsverbatim that of [FFS05, Proposition 5.2]. (cid:3) Proposition 5.2. The Chern-Weil homomorphism χ : S q ( h ∗ ) h → H q ( W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l , h ; K ) is an isomor-phism for q ≤ n − l . Furthermore, H q ( W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l , h , K ) = C q ( W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l , h ; K ) Proof. The injective Lie algebra homomorphism W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l ֒ → g induces a natural h -equivariantinjective map η : V q (cid:0) W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l / h (cid:1) → V q (cid:0) g / h (cid:1) which in turn gives an h -equivariant surjectivemorphism η ∗ : H q ( g , h ; K ) → H q ( W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l , h ; K ) . It is a straightforward verification that χ = η ∗ ◦ 𝒳 , where 𝒳 is the Chern-Weil homomorphism from Proposition 5.1. Thus, the map χ is surjective. Onthe other hand, on account of Corollary 1 in [Kho07], we have S q ( h ∗ ) h ∼ = H q ( W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l , h , K ) for q ≤ n − l . Hence, χ is in fact an isomorphism because S q ( h ∗ ) h is a finite-dimensional complex vector spacefor q ≤ n − l . We show that C q +1 ( W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l , h , K ) = 0 making use of invariant theory the sameway as in [Fuk86] which implies the second statement of the proposition. (cid:3) Let X ⊕ X ⊕ X ∈ h . Let ( ˆ A ~ Ch φ ~ Ch) k ∈ S ( h ∗ ) h be the homogeneous term of degree k in theTaylor expansion of ˆ A ~ Ch φ ~ Ch( X ) := ˆ A ~ ( X ) Ch φ ~ ( X ) Ch( X ) , where ˆ A ( X ) = det (cid:16) X/ X/ (cid:17) / and ˆ A ~ ( X ) = ˆ A ( ~ X ) , Ch φ ~ ( X ) = φ ~ (exp( X )) and Ch( X ) = tr(exp( X )) . Proposition 5.3. [ev Ψ n − l ] = ( − n − l 𝒳 (cid:0) ( ˆ A ~ Ch φ ~ Ch) n − l (cid:1) .Proof. By Proposition 5.1 there is an h -invariant polynomial P with 𝒳 ( P ) = [ev Ψ n − l ] . The ad -invarianceof P implies that it is uniquely determined by its value on the Cartan subalgebra a of h spanned by thefollowing group of vectors: x i ddx i , ≤ i ≤ n − l , E r a r a ⊗ , ≤ r a ≤ n a , ≤ a ≤ t , E rr , ≤ r ≤ N , where n a are the multiplicities of the t simple subrepresentations of C l ( cf. Lemma ?? ). Consider the commutativediagram S n − l ( h ∗ ) h H n − l ( g , h ; K ) C n − l ( W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l , h ; K ) χ 𝒳 η ∗ emanating from Proposition 5.2. Since all the arrows are isomorphisms here, we can prove the restriction ofthe desired identity to W (( ~ )) n − l ⋊ n ⊗ 𝒪 n − l , that is χ ( Q ) = η ∗ ([ev Ψ n − l ]) . The advantage of such a restrictionis that the identity becomes an identity of cocycles rather than of cohomology classes. To shorten notationthroughout the proof we shall write ev Ψ n − l to denote the cohomology class as well as its restriction tothe Lie subalgebra. Select an invariant polynomial P n − l whose restriction to a is given by P n − l ( M ⊗ a ⊗ b , . . . , M n − l ⊗ a n − l ⊗ b n − l ) = tr( M . . . M n − l ) φ ~ ( b . . . b n − l ) µ n − l Z [0 , n − l Y ≤ i ≤ j ≤ n − l exp( ~ ψ ( u i − u j ) α ij ) × ( a ⊗ · · · ⊗ a n ) du . . . du n − l . Here µ n − l , ψ, α ij , u , u n − l are exactly the same as in Section 2.3 in [FFS05]. We evaluate the cocycle ev Ψ n − l on the following special vectors u ij := − x i ddx i δ ij + x i x j ddx j v ir := x i ⊗ E rr w ir a := x i ⊗ E r a r a where the indices are as above. These vectors are in the kernel of pr and satisfy following commutatorrelations [ ddx i , u ij ] = x j ddx j [ ddx i , v ir ] = E rr [ ddx i , x i ⊗ E r a r a ] = E r a r a . Hence, C ( ddx i , u ij ) = − x j ddx j , C ( ddx i , v ir ) = − E rr and C ( ddx i , w ir a ) = − E r a r a . In what follows, we denoteby f i any vector of the form u ij with i ≥ j , or v ir or w ir a . Then χ ( P )( ddx ∧ f ∧ · · · ∧ ddx n − l ∧ f n − l ∧ ) = ( − n − l P ( df dx , . . . , df n − l dx n − l ) (37)where in χ ( P ) only permutations σ ∈ S n − l with σ ( i ) − σ ( i − 1) = 1 for all i = 2 , , . . . , n − l contributenon-trivially. The number of such permutations in S n − l equals the number of permutations of pairs oftuples (2 i − , i ) , i = 1 , . . . , n − l which is exactly ( n − l )! . The left hand side of (37) can be computed usingthe definition of ψ ~ n − l exactly the same way as in the proof of Theorem 5.1 in [FFS05] and that of Theorem3 in [RT12]. Namely, one gets χ ( P )( ddx ∧ f ∧ · · · ∧ ddx n − l ∧ f n − l ∧ ) = P n − l ( df dx , . . . , df n − l dx n − l ) which combined with 37 implies P = ( − n − l P n − l on a . Ergo, P = ( − n − l P n − l on h . It remains tocalculate P n − l on a . We start by remarking that P n = P ′ n φ ~ where P ′ n is the polynomial defined in Equation in the proof of [FFS05, Theorem 5.1]. In the same fashion as in [PPT07, Theorem 5.3] we explicitly calculate RACE DENSITIES AND ALGEBRAIC INDEX THEOREMS FOR THE SHEAF OF FORMAL CHEREDNIK ALGEBRAS 23 P n − l on the diagonal matrices X = Y + Z where Y := P n − li =1 ν i x i ddx + P Nr =1 σ r E rr ∈ gl n − l ( K ) ⊕ gl N ( K ) and Z := P ≤ a ≤ t, ≤ r a ≤ n a τ ar a E r a r a ∈ z ⊗ K . To that aim consider the generating function S ( X ) = P m ≥ m ! P m ( X, . . . , X ) . We then have S ( X ) = X l ! P ′ l ( Y, . . . , Y | {z } l times ) X k = m − l k ! φ ~ ( Z k ) = ( ˆ A ~ Ch)( Y ) Ch φ ~ ( Z ) . Since P n − l is the degree n − l component of S ( X ) , it is equal to (cid:0) ˆ A ~ Ch Ch φ ~ (cid:1) n − l . Hence ev Ψ n − l =( − n − l χ (cid:16)(cid:0) ˆ A ~ Ch Ch φ ~ (cid:1) n − l (cid:17) . The assertion follows. (cid:3) Let henceforth ϕ ∗ ∇ ∞ := ∇ + [ A, · ] be the flat smooth connection on the associated vector bun-dle F ( N ) × G 𝒜 H, (( ~ )) n − l,l ) over X Hi from Subsection 4.2 where ∇ is a smooth (non-flat) connection, A ∈ Ω ( X Hi , 𝒜 H, (( ~ )) n − l,l ) and G = GL n − l ( K ) × Z ⊗ K . Consequently, by defiintion ( ϕ ∗ ∇ ∞ ) = ∇ + [ ∇ A + 12 [ A, A ] , · ] = [Θ , · ] with a central element Θ ∈ Ω ( X Hi , K ) . At the same time the curvature of the non-flat connection ∇ canbe written in the form ∇ = [ R T + R N , · ] with R T ∈ Ω ( X Hi , gl n − l ( K )) and R N ∈ Ω ( X Hi , z ⊗ K ) fromwhich we conclude(38) ∇ A + 12 [ A, A ] = Θ − R T − R N . The following theorem and its proof mimic Theorem 6 from [RT12]. Theorem 5.4. The n − l -form χ Hi, (( ~ )) (1) − ~ n − l (cid:16) ˆ A ( R T ) Ch( − Θ ~ ) Ch φ ~ ( R N ~ ) (cid:17) n − l is an exact form on X Hi .Proof. We assume that A saturates pr( A ) = 0 . We are allowed to do this because ϕ ∗ ∇ ∞ can be rewritten inthe form ϕ ∗ ∇ ∞ = ∇ + [ A, − ] = ( ∇ + [pr( A ) , − ]) + [( A − pr( A )) , − ] = ˜ ∇ + [ ˜ A, − ] . Then, accounting that pr( ∇ A ) = ∇ pr( A ) for any pair of smooth vector fields ξ , ξ on X iH , we have C ( Aξ , Aξ ) = − pr( ∇ A ( ξ , ξ ) + [ A ( ξ ) , A ( ξ )]) = − pr(Θ − R T − R N ) = R T + R N − Θ where in the third equation we used (38).Denote the homogeneous h -invariant polynomial (cid:0) ˆ A ~ Ch Ch φ ~ (cid:1) n − l by P and let v , . . . , v n − l be vectorfields on X . Note that ev Ψ n − l ( A ⊗ n − l ) = (2 n − l )! ψ ~ n − l ( A ⊗ n − l ) . Then, in exactly the samefashion as in the proof of [RT12, Theorem 6] we have χ (( ~ )) ,iH (1)( v , . . . , v n − l ) = ( − n − l ev Ψ n − l ( A ∧ · · · ∧ A | {z } n − l times )( v , . . . , v n − l )= 𝒳 ( P )( A ∧ · · · ∧ A | {z } n − l times )( v , . . . , v n − l )= 1( n − l )! X σ ( − σ P ( C ( Av σ (1) , Av σ (2) ) , . . . , C ( Av σ (2 n − l − , Av σ (2 n − l ) ))= 1( n − l )! X σ ( − σ P (( R T + R N − Θ)( v σ (1) , v σ (2) ) , . . . , ( R T + R N − Θ)( v σ (2 n − l − , v σ (2 n − l ) ))= 1( n − l )! P (( R T + R N − Θ) , . . . , ( R T + R N − Θ) | {z } n − l times )( v , . . . , v n − l )= P ( R T + R N − Θ)( v , . . . , v n − l ) where in the definition of the trace density we implicitly used that ψ ~ n − l is gl n − l ( K ) ⊕ z ⊗ K -basic. Hence,modulo exact forms we have χ Hi, (( ~ )) (1) = (cid:0) ˆ A ~ ( R T ) Ch( − Θ) Ch φ ~ ( R N ) (cid:1) n − l = ~ n − l (cid:0) ˆ A ~ ( R T ~ ) Ch( − Θ ~ ) Ch φ ~ ( R N ~ ) (cid:1) n − l . The definition of ˆ A ~ and ˆ A imply the claim. (cid:3) Acknowledgemnts This work is based on part of my Ph.D thesis at the ETH Zurich. I am indebted to Prof. Giovanni Felderand Prof. Ajay Ramadoss for explaining to me in detail their previous works on related research matters andfor teaching me the techniques I applied in this note. References [AFLS00] J. Alev, M.A. Farinati, T. Lambre, and A.L. 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