Introduction
The ( Λ − α ) (\Lambda-\alpha) ( Λ − α ) g \mathfrak{g} g H ∗ ( M Λ , α ) H^*(\mathcal{M}_{\Lambda,\alpha}) H ∗ ( M Λ , α ) Z ( R α Λ ) Z(R^\Lambda_\alpha) Z ( R α Λ ) R α Λ R^\Lambda_\alpha R α Λ g \mathfrak{g} g Γ = ( I , E ) \Gamma=(I,E) Γ = ( I , E ) Λ \Lambda Λ α ∈ Q + \alpha\in Q^+ α ∈ Q +
Since the cyclotomic KLR algebra R α Λ R^\Lambda_\alpha R α Λ M Λ , α \mathcal{M}_{\Lambda,\alpha} M Λ , α [7 ] between cyclotomic KLR algebras and blocks of cyclotomic Hecke algebras intertwines this with the question of understanding the center of a cyclotomic Hecke algebra, a question of longstanding interest in representation theory.
The isomorphism ϕ : H ∗ ( M Λ , α ; C ) → Z ( R α Λ ( C ) ) \phi: H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{C}) \to Z(R^\Lambda_\alpha(\mathbb{C})) ϕ : H ∗ ( M Λ , α ; C ) → Z ( R α Λ ( C )) [6 ] and in general finite type by Shan, Varagnolo and Vasserot [32 ] and the fourth-named author [34 ] . The latter two proofs used techniques in the representation theory of current algebras which cannot be applied in affine type A.
The approach of [34 ] is greatly simplified by the work of McGerty and Nevins [24 ] on Kirwan surjectivity. The quiver varieties M Λ , α \mathcal{M}_{\Lambda,\alpha} M Λ , α V i V_i V i i ∈ I i \in I i ∈ I c i ; j ∈ H 2 j ( M Λ , α ; Z ) c_{i;j} \in H^{2j}(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z}) c i ; j ∈ H 2 j ( M Λ , α ; Z ) H ∗ ( M Λ , α ; Z ) H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z}) H ∗ ( M Λ , α ; Z )
Assume that α ∈ Q + \alpha\in Q^+ α ∈ Q + α \alpha α n n n i ∈ I , 1 ≤ j ≤ n i \in I, 1 \leq j \leq n i ∈ I , 1 ≤ j ≤ n
e i ; j : = ∑ ν ∈ I α e j ( y a ∣ ν a = i ) e ( ν ) ∈ Z ( R α Λ ) , e_{i;j}:=\sum_{\nu\in I^\alpha}e_j(y_a\mid\nu_a=i)e(\nu)\in Z(R^\Lambda_\alpha), e i ; j := ν ∈ I α ∑ e j ( y a ∣ ν a = i ) e ( ν ) ∈ Z ( R α Λ ) , where I α I^\alpha I α ν ∈ I n \nu\in I^n ν ∈ I n ∑ a α ν a = α \sum_a\alpha_{\nu_a}=\alpha ∑ a α ν a = α j j j i i i R α Λ R^\Lambda_\alpha R α Λ e i ; j e_{i;j} e i ; j i ∈ I , 1 ≤ j ≤ n i\in I,1\leq j\leq n i ∈ I , 1 ≤ j ≤ n Jucys–Murphy center , or JM center . The main result of this paper is the following:
Theorem A. Let Γ \Gamma Γ k \mathbb{k} k p ≥ 0 p\geq0 p ≥ 0
(1) There is an algebra homomorphism
ϕ k : H ∗ ( M Λ , α ; k ) → Z ( R α Λ ( k ) ) \phi_{\mathbb{k}}:H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{k})\to Z(R^\Lambda_\alpha(\mathbb{k})) ϕ k : H ∗ ( M Λ , α ; k ) → Z ( R α Λ ( k )) with ϕ k ( c i ; j ) = e i ; j \phi_{\mathbb{k}}(c_{i;j})=e_{i;j} ϕ k ( c i ; j ) = e i ; j i ∈ I , 1 ≤ j ≤ n i\in I,1\leq j\leq n i ∈ I , 1 ≤ j ≤ n R α Λ ( k ) R^\Lambda_\alpha(\mathbb{k}) R α Λ ( k )
(2) The map ϕ k \phi_{\mathbb{k}} ϕ k
dim H ∗ ( M Λ , α ; k ) ≥ dim Z ( R α Λ ( k ) ) . \dim H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{k})\geq\dim Z(R^\Lambda_\alpha(\mathbb{k})). dim H ∗ ( M Λ , α ; k ) ≥ dim Z ( R α Λ ( k )) . (3) If every connected component of Γ \Gamma Γ e e e e e e p = 0 p=0 p = 0 p ∤ e p\nmid e p ∤ e p = e p=e p = e ϕ k \phi_{\mathbb{k}} ϕ k
The conditions on p p p p = 0 p=0 p = 0 p ∤ e p\nmid e p ∤ e Theorem 4.5 . If p = e p=e p = e [6 ] . This gives the upper bound required in part (2).
Let H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q G ( r , 1 , n ) G(r,1,n) G ( r , 1 , n ) k \mathbb{k} k q ≠ 1 q\neq1 q = 1 Q = ( Q 1 , … , Q r ) ∈ ( k × ) r \mathbf{Q}=(Q_1,\ldots,Q_r)\in(\mathbb{k}^{\times})^r Q = ( Q 1 , … , Q r ) ∈ ( k × ) r Definition 4.1 . It is a well-known result of Jucys [18 (8)] and Murphy [25 Theorem 1.9] that the center of the group algebra of the symmetric group is the set of symmetric polynomials in the Jucys–Murphy elements of this algebra. Brundan [6 Theorem 1] proved the corresponding result for the degenerate Ariki–Koike algebras. Hence, it is natural to make the following conjecture, which was part of the folklore long before it first appeared in the literature in [23 ] .
Conjecture 1.1 (Center conjecture [23 Conjecture 3.1] ). Suppose that q − 1 ∈ k × q-1\in\mathbb{k}^{\times} q − 1 ∈ k × H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q L 1 , … , L n L_1,\ldots,L_n L 1 , … , L n
Equivalently, Conjecture 1.1 says that the center of the affine Hecke algebra of type A surjects onto the center of the Ariki–Koike algebras. In the semisimple case, Conjecture 1.1 was proved by Ariki and Koike [3 Theorem 3.20] . McGerty [23 Theorem 2.5] established the conjecture when n = 2 n=2 n = 2 r = 1 r=1 r = 1 [11 ] . Importantly, Ariki [2 §2, Example] gives an example to show that the assumption q ≠ 1 q\neq1 q = 1
Together with the isomorphism of Brundan and Kleshchev [7 ] , Theorem A(3) gives the center conjecture for the Ariki–Koike algebras:
Theorem B. Suppose that k \mathbb{k} k q ≠ 1 q\neq1 q = 1 H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q L 1 , … , L n L_1,\ldots,L_n L 1 , … , L n
Since Theorem B holds in every characteristic, we can apply Theorem B to prove a generalization of a conjecture of Krylov, Paegelow and Shlykov [21 ] that relates the equivariant K K K G ( n , r ) \mathcal{G}(n,r) G ( n , r ) L ( n , r ) \mathcal{L}(n,r) L ( n , r ) A ∘ = Z [ q ± 1 , Q 1 ± 1 , … , Q r ± 1 , ( q − 1 ) − 1 ] \mathcal{A}^{\circ}=\mathbb{Z}[q^{\pm1},Q_{1}^{\pm1},\ldots,Q_{r}^{\pm1},(q-1)^{-1}] A ∘ = Z [ q ± 1 , Q 1 ± 1 , … , Q r ± 1 , ( q − 1 ) − 1 ] § 5 for the unexplained notation below.
Theorem C.
(1) Let k \mathbb{k} k q , Q 1 , … , Q r , q − 1 ∈ k × q,Q_{1},\ldots,Q_{r},q-1\in\mathbb{k}^{\times} q , Q 1 , … , Q r , q − 1 ∈ k ×
Z ( H q , n Q ( k ) ) = Z ( H q , n Q ( k ) ) J M . Z(\mathcal{H}^{\mathbf{Q}}_{q,n}(\mathbb{k}))=Z(\mathcal{H}^{\mathbf{Q}}_{q,n}(\mathbb{k}))^{JM}. Z ( H q , n Q ( k )) = Z ( H q , n Q ( k ) ) J M . In particular, this holds over A ∘ \mathcal{A}^{\circ} A ∘
(2) There are compatible isomorphisms:
A ∘ ⊗ K T ( G ( n , r ) ) ≅ Z ( H q , n Q ( A ∘ ) ) and A ∘ ⊗ K T ( L ( n , r ) ) ≅ Tr ( H q , n Q ( A ∘ ) ) . \mathcal{A}^{\circ}\otimes K^{T}(\mathcal{G}(n,r))\cong Z(\mathcal{H}^{\mathbf{Q}}_{q,n}(\mathcal{A}^{\circ}))\qquad\text{and}\qquad\mathcal{A}^{\circ}\otimes K^{T}(\mathcal{L}(n,r))\cong\operatorname{Tr}(\mathcal{H}^{\mathbf{Q}}_{q,n}(\mathcal{A}^{\circ})). A ∘ ⊗ K T ( G ( n , r )) ≅ Z ( H q , n Q ( A ∘ )) and A ∘ ⊗ K T ( L ( n , r )) ≅ Tr ( H q , n Q ( A ∘ )) . (3) Let e ≥ 2 e\geq2 e ≥ 2 k \mathbb{k} k e e e ζ ∈ k \zeta\in\mathbb{k} ζ ∈ k e e e κ 1 , … , κ r ∈ Z \kappa_{1},\ldots,\kappa_{r}\in\mathbb{Z} κ 1 , … , κ r ∈ Z Λ = ∑ m Λ κ ‾ m \Lambda=\sum_{m}\Lambda_{\overline{\kappa}_{m}} Λ = ∑ m Λ κ m A e − 1 ( 1 ) A^{(1)}_{e-1} A e − 1 ( 1 ) α ∈ Q + ∨ \alpha\in Q_{+}^{\vee} α ∈ Q + ∨ k \mathbb{k} k K ( M Λ , α ; k ) ≅ Z ( H ζ , α Λ ( k ) ) K(\mathcal{M}_{\Lambda,\alpha};\mathbb{k})\cong Z(\mathcal{H}^{\Lambda}_{\zeta,\alpha}(\mathbb{k})) K ( M Λ , α ; k ) ≅ Z ( H ζ , α Λ ( k )) K K K k \mathbb{k} k q = ζ q=\zeta q = ζ Q m = ζ κ m Q_{m}=\zeta^{\kappa_{m}} Q m = ζ κ m
For k = C \mathbb{k}=\mathbb{C} k = C [21 Th. 5.4] , which was proved assuming [21 Conj. 4.6] ; part (3) removes this assumption. The matching of the components of the fixed locus with blocks of the Hecke algebra follows Paegelow [29 Th. 2.15, Prop. 2.17 and Prop. 5.3] .
The key new ingredient is injectivity over an arbitrary field of the map from cohomology to Z ( R α Λ ) Z(R_{\alpha}^{\Lambda}) Z ( R α Λ ) [34 ] : ϕ \phi ϕ [27 Prop. 12.3.2, 13.3.1] to guarantee that H ∗ ( M Λ ) = ⨁ α H ∗ ( M Λ , α ) H^{*}(\mathcal{M}_{\Lambda})=\bigoplus_{\alpha}H^{*}(\mathcal{M}_{\Lambda,\alpha}) H ∗ ( M Λ ) = ⨁ α H ∗ ( M Λ , α ) H ∗ ( M Λ , 0 ) ≅ k H^{*}(\mathcal{M}_{\Lambda,0})\cong\mathbb{k} H ∗ ( M Λ , 0 ) ≅ k L Λ , α = π − 1 ( 0 ) ⊂ M Λ , α \mathcal{L}_{\Lambda,\alpha}=\pi^{-1}(0)\subset\mathcal{M}_{\Lambda,\alpha} L Λ , α = π − 1 ( 0 ) ⊂ M Λ , α H ∗ ( M Λ , 0 ) ≅ k H_{*}(\mathcal{M}_{\Lambda,0})\cong\mathbb{k} H ∗ ( M Λ , 0 ) ≅ k
Thus, some explanation is needed about how our results here are different from [34 ] :
Since [34 ] was written when Kirwan surjectivity was not known, it used the interpretation of H ∗ ( M Λ , α ) H^{*}(\mathcal{M}_{\Lambda,\alpha}) H ∗ ( M Λ , α ) M Λ , α \mathcal{M}_{\Lambda,\alpha} M Λ , α ϕ \phi ϕ [34 ] to verify that their relations are preserved.
For reduction to positive characteristic, we need a more careful analysis of the geometry of quiver varieties and Hecke correspondences, resulting in the stronger integral statement proved in Theorem 2.8: the Chow groups of the cores are generated over Z \mathbb{Z} Z
The proof of Theorem A(1)–(2) has three steps. First, in § 2 , we define decorated divided-power Hecke correspondences on the Chow groups of the cores of quiver varieties and prove integral cyclicity (Theorem 2.8). Next, in § 3 , we transpose these to raising and lowering operators on integral cohomology, identify the integral form of the loop algebra they generate, and compare its action with the corresponding operators on the KLR side. Finally, § 3.4 uses Kirwan surjectivity to construct the integral comparison map and cyclicity to prove its injectivity after arbitrary base change. Part (3) and Theorem B are proved in § 4 . Theorem C is proved in § 5 .
Quiver varieties and Chow groups
Notation
2.1. Notation. We follow the conventions of [27 ] . Fix Γ = ( I , E ) \Gamma=(I,E) Γ = ( I , E ) v = ( v i ) i ∈ I , w = ( w i ) i ∈ I ∈ Z ≥ 0 I \mathbf{v}=(v_{i})_{i\in I},\mathbf{w}=(w_{i})_{i\in I}\in\mathbb{Z}_{\geq0}^{I} v = ( v i ) i ∈ I , w = ( w i ) i ∈ I ∈ Z ≥ 0 I k ∈ I k\in I k ∈ I e k ∈ Z ≥ 0 I \mathbf{e}_{k}\in\mathbb{Z}_{\geq0}^{I} e k ∈ Z ≥ 0 I k k k M ( v , w ) \mathcal{M}(\mathbf{v},\mathbf{w}) M ( v , w ) M 0 ( v , w ) \mathcal{M}_{0}(\mathbf{v},\mathbf{w}) M 0 ( v , w ) [27 (2.2.2), (2.2.1)] , π : M ( v , w ) → M 0 ( v , w ) \pi:\mathcal{M}(\mathbf{v},\mathbf{w})\to\mathcal{M}_{0}(\mathbf{v},\mathbf{w}) π : M ( v , w ) → M 0 ( v , w ) Λ = ∑ i w i Λ i \Lambda= \sum_i w_i\Lambda_i Λ = ∑ i w i Λ i α = ∑ i v i α i ∈ Q + \alpha= \sum_i v_i\alpha_i \in Q^+ α = ∑ i v i α i ∈ Q + μ = Λ − α \mu= \Lambda- \alpha μ = Λ − α
M ( v , w ) = M Λ , α , L ( v , w ) = π − 1 ( 0 ) = L Λ , α , M Λ = ∐ α ∈ Q + M Λ , α , L Λ = ∐ α ∈ Q + L Λ , α . \mathcal{M}(\mathbf{v},\mathbf{w}) = \mathcal{M}_{\Lambda,\alpha}, \qquad\mathcal{L}(\mathbf{v},\mathbf{w}) = \pi^{-1}(0) = \mathcal{L}_{\Lambda,\alpha}, \qquad\mathcal{M}_{\Lambda} = \coprod_{\alpha\in Q^+} \mathcal{M}_{\Lambda,\alpha}, \qquad\mathcal{L}_{\Lambda} = \coprod_{\alpha\in Q^+} \mathcal{L}_{\Lambda,\alpha}. M ( v , w ) = M Λ , α , L ( v , w ) = π − 1 ( 0 ) = L Λ , α , M Λ = α ∈ Q + ∐ M Λ , α , L Λ = α ∈ Q + ∐ L Λ , α . For n ≥ 0 n \ge0 n ≥ 0
M Λ , n = ∐ α ∈ Q n + M Λ , α . \mathcal{M}_{\Lambda,n} = \coprod_{\alpha\in Q^+_n} \mathcal{M}_{\Lambda,\alpha}. M Λ , n = α ∈ Q n + ∐ M Λ , α . Here Q n + Q^+_n Q n + α = ∑ i v i α i ∈ Q + \alpha= \sum_i v_i\alpha_i \in Q^+ α = ∑ i v i α i ∈ Q + ∑ i v i = n \sum_i v_i = n ∑ i v i = n
H ∗ ( M Λ ) = ⨁ α H ∗ ( M Λ , α ) = ⨁ n H ∗ ( M Λ , n ) , H^*(\mathcal{M}_{\Lambda}) = \bigoplus_{\alpha} H^*(\mathcal{M}_{\Lambda,\alpha}) = \bigoplus_n H^*(\mathcal{M}_{\Lambda,n}), H ∗ ( M Λ ) = α ⨁ H ∗ ( M Λ , α ) = n ⨁ H ∗ ( M Λ , n ) , and similarly for L Λ \mathcal{L}_{\Lambda} L Λ
Throughout, we will consider varieties X X X C \mathbb{C} C A ∗ ( X ) A_*(X) A ∗ ( X ) X X X H ∗ ( − ) H_*(-) H ∗ ( − ) H ∗ o r d ( − ) H_*^{\mathrm{ord}}(-) H ∗ ord ( − ) L ( v , w ) \mathcal{L}(\mathbf{v},\mathbf{w}) L ( v , w ) H ∗ ( − ) H_*(-) H ∗ ( − ) H ∗ o r d ( − ) H_*^{\mathrm{ord}}(-) H ∗ ord ( − )
Lemma 2.1.
M ( v , w ) \mathcal{M}(\mathbf{v},\mathbf{w}) M ( v , w ) L ( v , w ) \mathcal{L}(\mathbf{v},\mathbf{w}) L ( v , w ) M ( v , w ) \mathcal{M}(\mathbf{v},\mathbf{w}) M ( v , w ) [27 Prop. 2.3.8] .
The varieties M ( v , w ) \mathcal{M}(\mathbf{v},\mathbf{w}) M ( v , w ) L ( v , w ) \mathcal{L}(\mathbf{v},\mathbf{w}) L ( v , w ) [27 §7.1] : their odd homology vanishes, their even homology is a free abelian group, and the cycle map cl : A ∗ → H 2 ∗ \operatorname{cl}: A_* \to H_{2*} cl : A ∗ → H 2 ∗ [27 Th. 7.3.5] .
For any field k \mathbb{k} k
H ∗ ( M ( v , w ) ; k ) = H ∗ ( M ( v , w ) ; Z ) ⊗ k , H ∗ ( L ( v , w ) ; Z ) ≅ H ∗ o r d ( M ( v , w ) ; Z ) (1) H^*(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{k}) = H^*(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{Z}) \otimes\mathbb{k}, \qquad H_*(\mathcal{L}(\mathbf{v},\mathbf{w});\mathbb{Z}) \cong H_*^{\mathrm{ord}}(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{Z}) \tag*{(1)} H ∗ ( M ( v , w ) ; k ) = H ∗ ( M ( v , w ) ; Z ) ⊗ k , H ∗ ( L ( v , w ) ; Z ) ≅ H ∗ ord ( M ( v , w ) ; Z ) (1) and the induced pairing
H k ( M ( v , w ) ; k ) ⊗ H k ( L ( v , w ) ; k ) ⟶ k , m ⊗ c ↦ ⟨ m ∣ L ( v , w ) , c ⟩ (2) H^k(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{k}) \otimes H_k(\mathcal{L}(\mathbf{v},\mathbf{w});\mathbb{k}) \longrightarrow\mathbb{k}, \qquad m \otimes c \mapsto\langle m|_{\mathcal{L}(\mathbf{v},\mathbf{w})},c\rangle \tag*{(2)} H k ( M ( v , w ) ; k ) ⊗ H k ( L ( v , w ) ; k ) ⟶ k , m ⊗ c ↦ ⟨ m ∣ L ( v , w ) , c ⟩ (2) is perfect.
Proof. Parts (1) and (2) are cited. For (3), since the inclusion L = L ( v , w ) ↪ M ( v , w ) \mathcal{L} = \mathcal{L}(\mathbf{v},\mathbf{w}) \hookrightarrow\mathcal{M}(\mathbf{v},\mathbf{w}) L = L ( v , w ) ↪ M ( v , w ) L ( v , w ) \mathcal{L}(\mathbf{v},\mathbf{w}) L ( v , w ) H ∗ ( L ( v , w ) ; Z ) ≅ H ∗ o r d ( L ( v , w ) ; Z ) ≅ H ∗ o r d ( M ( v , w ) ; Z ) H_*(\mathcal{L}(\mathbf{v},\mathbf{w});\mathbb{Z}) \cong H_*^{\mathrm{ord}}(\mathcal{L}(\mathbf{v},\mathbf{w});\mathbb{Z}) \cong H_*^{\mathrm{ord}}(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{Z}) H ∗ ( L ( v , w ) ; Z ) ≅ H ∗ ord ( L ( v , w ) ; Z ) ≅ H ∗ ord ( M ( v , w ) ; Z ) □ \square □
Hecke correspondences
Following the conventions of Nakajima, we represent a point in M ( v , w ) \mathcal{M}(\mathbf{v},\mathbf{w}) M ( v , w ) x = [ B , i , j ] x = [B,i,j] x = [ B , i , j ] G v G_{\mathbf{v}} G v k ∈ I k \in I k ∈ I n ≥ 1 n \ge1 n ≥ 1 v k ≥ n v_k \ge n v k ≥ n
We will make heavy use of the Hecke correspondence
P k ( n ) = P k ( n ) ( v , w ) ⊂ M ( v − n e k , w ) × M ( v , w ) \mathfrak{P}^{(n)}_k = \mathfrak{P}^{(n)}_k(\mathbf{v},\mathbf{w}) \subset\mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) \times\mathcal{M}(\mathbf{v},\mathbf{w}) P k ( n ) = P k ( n ) ( v , w ) ⊂ M ( v − n e k , w ) × M ( v , w ) of [27 §5.3] . If we regard M ( v , w ) \mathcal{M}(\mathbf{v},\mathbf{w}) M ( v , w ) P k ( n ) \mathfrak{P}^{(n)}_k P k ( n ) V 1 ⊂ V 2 V^1 \subset V^2 V 1 ⊂ V 2 V 2 V^2 V 2 v \mathbf{v} v V 1 V^1 V 1 v − n e k \mathbf{v}-n\mathbf{e}_k v − n e k i i i [27 (5.3.1)] , with the projections p 1 p_1 p 1 p 2 p_2 p 2 V 1 V^1 V 1 V 2 V^2 V 2 p 1 , p 2 p_1,p_2 p 1 , p 2
An important role in understanding these fibers is played by the complex of [27 (2.9.1)]
V k → σ k ⨁ in ( h ) = k V out ( h ) ⊕ W k → τ k V k , σ k = ⨁ in ( h ) = k B h ˉ ⊕ j k , τ k = ∑ in ( h ) = k ϵ ( h ) B h + i k . V_k \xrightarrow{\sigma_k} \bigoplus_{\operatorname{in}(h)=k} V_{\operatorname{out}(h)} \oplus W_k \xrightarrow{\tau_k} V_k, \qquad\sigma_k = \bigoplus_{\operatorname{in}(h)=k} B_{\bar h} \oplus j_k, \qquad\tau_k = \sum_{\operatorname{in}(h)=k} \epsilon(h)B_h+i_k. V k σ k in ( h ) = k ⨁ V out ( h ) ⊕ W k τ k V k , σ k = in ( h ) = k ⨁ B h ˉ ⊕ j k , τ k = in ( h ) = k ∑ ϵ ( h ) B h + i k . where h ˉ \bar h h ˉ h h h ϵ \epsilon ϵ [27 §2.1] . Thus τ k \tau_k τ k V k V_k V k σ k \sigma_k σ k τ k σ k \tau_k\sigma_k τ k σ k k k k σ k \sigma_k σ k p 2 p_2 p 2 V 2 V^2 V 2 V k 2 V^2_k V k 2 v k − n v_k-n v k − n τ k \tau_k τ k p 1 p_1 p 1 V 1 V^1 V 1 M ( v − n e k , w ) \mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) M ( v − n e k , w ) n n n ker τ k / im σ k \ker\tau_k/\operatorname{im}\sigma_k ker τ k / im σ k σ k \sigma_k σ k τ k \tau_k τ k V 1 V^{1} V 1 V 1 V^{1} V 1 V 2 V^{2} V 2 k k k τ k \tau_k τ k σ k \sigma_k σ k V 2 V^{2} V 2 V k 2 V^{2}_k V k 2 ker τ k \ker\tau_k ker τ k im ( σ k ) \operatorname{im}(\sigma_k) im ( σ k ) dim im ( σ k ) + n \dim\operatorname{im}(\sigma_k)+n dim im ( σ k ) + n V 2 ⊃ V 1 V^{2} \supset V^{1} V 2 ⊃ V 1
Let S S S n n n V k 2 / V k 1 V^{2}_k/V^{1}_k V k 2 / V k 1 P k ( n ) \mathfrak{P}^{(n)}_k P k ( n ) Sym n = Z [ x 1 , … , x n ] S n ≅ Z [ c 1 , … , c n ] \operatorname{Sym}_n = \mathbb{Z}[x_1,\ldots,x_n]^{S_n} \cong\mathbb{Z}[c_1,\ldots,c_n] Sym n = Z [ x 1 , … , x n ] S n ≅ Z [ c 1 , … , c n ] P k ( n ) \mathfrak{P}^{(n)}_k P k ( n ) c j c_j c j c j ( S ) c_j(S) c j ( S )
Definition 2.2. For f ∈ Sym n f \in\operatorname{Sym}_n f ∈ Sym n
F k ( n ) [ f ] : A ∗ ( L ( v − n e k , w ) ) ⟶ A ∗ ( L ( v , w ) ) , F k ( n ) [ f ] ( c ) = q ∗ δ ′ ( c × ( f ∩ [ P k ( n ) ] ) ) . F_k^{(n)}[f]\colon A_*(\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})) \longrightarrow A_*(\mathcal{L}(\mathbf{v},\mathbf{w})), \qquad F_k^{(n)}[f](c)=q_*\delta'(c \mathbin{\times}(f\cap[\mathfrak{P}^{(n)}_k])). F k ( n ) [ f ] : A ∗ ( L ( v − n e k , w )) ⟶ A ∗ ( L ( v , w )) , F k ( n ) [ f ] ( c ) = q ∗ δ ′ ( c × ( f ∩ [ P k ( n ) ])) . Here δ \delta δ M ( v − n e k , w ) \mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) M ( v − n e k , w ) δ ′ \delta' δ ′ [13 §6.2] for the fiber square
L ( v − n e k , w ) × M ( v − n e k , w ) P k ( n ) → L ( v − n e k , w ) × P k ( n ) pr 1 ↓ ↓ incl × p 1 M ( v − n e k , w ) → δ M ( v − n e k , w ) × M ( v − n e k , w ) \begin{CD}
\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})\mathbin{\times}_{\mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})}\mathfrak{P}^{(n)}_k @>>> \mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})\mathbin{\times}\mathfrak{P}^{(n)}_k \\
@V{\operatorname{pr}_1}VV @VV{\operatorname{incl}\times p_1}V \\
\mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) @>{\delta}>> \mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})\mathbin{\times}\mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})
\end{CD} L ( v − n e k , w ) × M ( v − n e k , w ) P k ( n ) pr 1 ↓ ⏐ M ( v − n e k , w ) δ L ( v − n e k , w ) × P k ( n ) ↓ ⏐ incl × p 1 M ( v − n e k , w ) × M ( v − n e k , w ) and q : L ( v − n e k , w ) × M ( v − n e k , w ) P k ( n ) → L ( v , w ) q\colon\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})\mathbin{\times}_{\mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})}\mathfrak{P}^{(n)}_k \to\mathcal{L}(\mathbf{v},\mathbf{w}) q : L ( v − n e k , w ) × M ( v − n e k , w ) P k ( n ) → L ( v , w ) p 2 p_2 p 2 pr 1 \operatorname{pr}_1 pr 1 p 1 p_1 p 1
Similarly, we have a decorated divided-power raising operator
E k ( n ) [ f ] : A ∗ ( L ( v , w ) ) ⟶ A ∗ ( L ( v − n e k , w ) ) E_k^{(n)}[f]\colon A_*(\mathcal{L}(\mathbf{v},\mathbf{w})) \longrightarrow A_*(\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})) E k ( n ) [ f ] : A ∗ ( L ( v , w )) ⟶ A ∗ ( L ( v − n e k , w )) on Chow groups, defined by the same formula with the roles of p 1 p_1 p 1 p 2 p_2 p 2
The same formula in Borel–Moore homology defines operators F k ( n ) [ f ] , E k ( n ) [ f ] F_k^{(n)}[f], E_k^{(n)}[f] F k ( n ) [ f ] , E k ( n ) [ f ] H ∗ ( L ; Z ) H_*(\mathcal{L};\mathbb{Z}) H ∗ ( L ; Z ) f ∩ [ P k ( n ) ] f\cap[\mathfrak{P}^{(n)}_k] f ∩ [ P k ( n ) ] [8 §2.7] :
F k ( n ) [ f ] : H ∗ ( L ( v − n e k , w ) ) ⟶ H ∗ ( L ( v , w ) ) , F k ( n ) [ f ] ( c ) = ( p 2 ) ∗ ( p 1 ∗ ( c ) ∩ ( f ∩ [ P k ( n ) ] ) ) , E k ( n ) [ f ] : H ∗ ( L ( v , w ) ) ⟶ H ∗ ( L ( v − n e k , w ) ) , E k ( n ) [ f ] ( c ) = ( p 1 ) ∗ ( p 2 ∗ ( c ) ∩ ( f ∩ [ P k ( n ) ] ) ) . (3) \begin{aligned}
F_k^{(n)}[f]\colon H_*(\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})) &\longrightarrow H_*(\mathcal{L}(\mathbf{v},\mathbf{w})), & F_k^{(n)}[f](c)&=(p_2)_*(p_1^*(c)\cap(f\cap[\mathfrak{P}^{(n)}_k])),\\
E_k^{(n)}[f]\colon H_*(\mathcal{L}(\mathbf{v},\mathbf{w})) &\longrightarrow H_*(\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})), & E_k^{(n)}[f](c)&=(p_1)_*(p_2^*(c)\cap(f\cap[\mathfrak{P}^{(n)}_k])). \tag*{(3)}
\end{aligned} F k ( n ) [ f ] : H ∗ ( L ( v − n e k , w )) E k ( n ) [ f ] : H ∗ ( L ( v , w )) ⟶ H ∗ ( L ( v , w )) , ⟶ H ∗ ( L ( v − n e k , w )) , F k ( n ) [ f ] ( c ) E k ( n ) [ f ] ( c ) = ( p 2 ) ∗ ( p 1 ∗ ( c ) ∩ ( f ∩ [ P k ( n ) ])) , = ( p 1 ) ∗ ( p 2 ∗ ( c ) ∩ ( f ∩ [ P k ( n ) ])) . (3) Since the cycle map commutes with exterior products, proper pushforward, refined Gysin maps and Chern classes [13 Ch. 19] , the cycle map cl : A ∗ ( L ) → H ∗ ( L ) \operatorname{cl}\colon A_*(\mathcal{L})\to H_*(\mathcal{L}) cl : A ∗ ( L ) → H ∗ ( L )
On cohomology, we define the corresponding operators by pull-push: for f ∈ Sym n f\in\operatorname{Sym}_n f ∈ Sym n
F k ( n ) [ f ] : H ∗ ( M ( v − n e k , w ) ) ⟶ H ∗ ( M ( v , w ) ) , F k ( n ) [ f ] ( m ) = ( p 2 ) ∗ ( f ( S ) ∪ p 1 ∗ m ) , E k ( n ) [ f ] : H ∗ ( M ( v , w ) ) ⟶ H ∗ ( M ( v − n e k , w ) ) , E k ( n ) [ f ] ( m ) = ( p 1 ) ∗ ( f ( S ) ∪ p 2 ∗ m ) , (4) \begin{aligned}
F_k^{(n)}[f]\colon H^*(\mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})) &\longrightarrow H^*(\mathcal{M}(\mathbf{v},\mathbf{w})), & F_k^{(n)}[f](m)&=(p_2)_*(f(S)\cup p_1^*m),\\
E_k^{(n)}[f]\colon H^*(\mathcal{M}(\mathbf{v},\mathbf{w})) &\longrightarrow H^*(\mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})), & E_k^{(n)}[f](m)&=(p_1)_*(f(S)\cup p_2^*m), \tag*{(4)}
\end{aligned} F k ( n ) [ f ] : H ∗ ( M ( v − n e k , w )) E k ( n ) [ f ] : H ∗ ( M ( v , w )) ⟶ H ∗ ( M ( v , w )) , ⟶ H ∗ ( M ( v − n e k , w )) , F k ( n ) [ f ] ( m ) E k ( n ) [ f ] ( m ) = ( p 2 ) ∗ ( f ( S ) ∪ p 1 ∗ m ) , = ( p 1 ) ∗ ( f ( S ) ∪ p 2 ∗ m ) , (4) where ( p 1 ) ∗ (p_1)_* ( p 1 ) ∗ ( p 2 ) ∗ (p_2)_* ( p 2 ) ∗ p 1 , p 2 p_1,p_2 p 1 , p 2 P k ( n ) \mathfrak{P}^{(n)}_k P k ( n ) M ( v − n e k , w ) \mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) M ( v − n e k , w ) M ( v , w ) \mathcal{M}(\mathbf{v},\mathbf{w}) M ( v , w ) Definition 2.2 on the Borel–Moore homology of M \mathcal{M} M E k , r = E k ( 1 ) [ x r ] E_{k,r}=E_k^{(1)}[x^r] E k , r = E k ( 1 ) [ x r ] F k , r = F k ( 1 ) [ x r ] F_{k,r}=F_k^{(1)}[x^r] F k , r = F k ( 1 ) [ x r ]
Lemma 2.3. Under the perfect pairing of Lemma 2.1 , the operator F k ( n ) [ f ] F_k^{(n)}[f] F k ( n ) [ f ] H ∗ ( L ) H_*(\mathcal{L}) H ∗ ( L ) E k ( n ) [ f ] E_k^{(n)}[f] E k ( n ) [ f ] H ∗ ( M ) H^*(\mathcal{M}) H ∗ ( M ) (4) , and similarly with E E E F F F
Proof. Let c ∈ H ∗ ( L ( v − n e k , w ) ) c\in H_*(\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})) c ∈ H ∗ ( L ( v − n e k , w )) m ∈ H ∗ ( M ( v , w ) ) m\in H^*(\mathcal{M}(\mathbf{v},\mathbf{w})) m ∈ H ∗ ( M ( v , w )) γ = f ∩ [ P k ( n ) ] \gamma=f\cap[\mathfrak{P}^{(n)}_k] γ = f ∩ [ P k ( n ) ] [8 §§2.6–2.7] give
⟨ m , F k ( n ) [ f ] ( c ) ⟩ = deg ( m ∩ q ∗ δ ′ ( c × γ ) ) = deg ( q ∗ m ∩ δ ′ ( c × γ ) ) = deg δ ′ ( c × ( p 2 ∗ m ∩ γ ) ) = ⟨ ( p 1 ) ∗ ( f ( S ) ∪ p 2 ∗ m ) , c ⟩ = ⟨ E k ( n ) [ f ] ( m ) , c ⟩ . \begin{aligned}
\langle m,F_k^{(n)}[f](c)\rangle&= \deg\left(m\cap q_*\delta'(c\times\gamma)\right)=\deg\left(q^*m\cap\delta'(c\times\gamma)\right)\\
&=\deg\delta'(c\times(p_2^*m\cap\gamma))=\langle(p_1)_*(f(S)\cup p_2^*m),c\rangle\\
&=\langle E_k^{(n)}[f](m),c\rangle.
\end{aligned} ⟨ m , F k ( n ) [ f ] ( c )⟩ = deg ( m ∩ q ∗ δ ′ ( c × γ ) ) = deg ( q ∗ m ∩ δ ′ ( c × γ ) ) = deg δ ′ ( c × ( p 2 ∗ m ∩ γ )) = ⟨( p 1 ) ∗ ( f ( S ) ∪ p 2 ∗ m ) , c ⟩ = ⟨ E k ( n ) [ f ] ( m ) , c ⟩ . The statement with E E E F F F
The stratification by epsilon k
As in the previous subsection, we keep k ∈ I k \in I k ∈ I x = [ B , i , j ] ∈ M ( v , w ) x=[B,i,j]\in\mathcal{M}(\mathbf{v},\mathbf{w}) x = [ B , i , j ] ∈ M ( v , w ) ϵ k ( x ) = codim V k Im τ k \epsilon_k(x)=\operatorname{codim}_{V_k}\operatorname{Im}\tau_k ϵ k ( x ) = codim V k Im τ k [27 §5.4] , let
M k ; n ( v , w ) = { x ∈ M ( v , w ) ∣ ϵ k ( x ) = n } , M k ; ≤ n = ⋃ m ≤ n M k ; m , M k ; ≥ n = ⋃ m ≥ n M k ; m . \mathcal{M}_{k;n}(\mathbf{v},\mathbf{w})=\{x\in\mathcal{M}(\mathbf{v},\mathbf{w})\mid\epsilon_k(x)=n\},\qquad\mathcal{M}_{k;\leq n}=\bigcup_{m\leq n}\mathcal{M}_{k;m},\qquad\mathcal{M}_{k;\geq n}=\bigcup_{m\geq n}\mathcal{M}_{k;m}. M k ; n ( v , w ) = { x ∈ M ( v , w ) ∣ ϵ k ( x ) = n } , M k ; ≤ n = m ≤ n ⋃ M k ; m , M k ; ≥ n = m ≥ n ⋃ M k ; m . and similarly L k ; n = L ∩ M k ; n \mathcal{L}_{k;n}=\mathcal{L}\cap\mathcal{M}_{k;n} L k ; n = L ∩ M k ; n M k ; ≤ n \mathcal{M}_{k;\leq n} M k ; ≤ n M k ; ≥ n \mathcal{M}_{k;\geq n} M k ; ≥ n M k ; n \mathcal{M}_{k;n} M k ; n M k ; ≤ n \mathcal{M}_{k;\leq n} M k ; ≤ n Y ⊂ M ( v , w ) Y\subset\mathcal{M}(\mathbf{v},\mathbf{w}) Y ⊂ M ( v , w ) ϵ k \epsilon_k ϵ k Y Y Y ϵ k ( Y ) \epsilon_k(Y) ϵ k ( Y ) Y ⊂ M k ; ≥ ϵ k ( Y ) Y\subset\mathcal{M}_{k;\geq\epsilon_k(Y)} Y ⊂ M k ; ≥ ϵ k ( Y )
This filtration is a geometric avatar of decomposing a representation according to the action of the s l 2 \mathfrak{sl}_2 sl 2 ± α k \pm\alpha_k ± α k M k ; 0 \mathcal{M}_{k;0} M k ; 0
Write M ( v − n e k , w ) ∘ = M k ; 0 ( v − n e k , w ) \mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})^\circ=\mathcal{M}_{k;0}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) M ( v − n e k , w ) ∘ = M k ; 0 ( v − n e k , w ) M k ; n ( v , w ) \mathcal{M}_{k;n}(\mathbf{v},\mathbf{w}) M k ; n ( v , w ) v − n e k \mathbf{v}-n\mathbf{e}_k v − n e k p 2 p_2 p 2 Q Q Q M ( v − n e k , w ) ∘ \mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})^\circ M ( v − n e k , w ) ∘ Q k ( v − n e k , w ) = ker τ k / Im σ k Q_k(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})=\ker\tau_k/\operatorname{Im}\sigma_k Q k ( v − n e k , w ) = ker τ k / Im σ k [27 (2.9.1)] , a vector bundle of rank m = ⟨ α k ∨ , μ ⟩ + 2 n m=\langle\alpha_k^\vee,\mu\rangle+2n m = ⟨ α k ∨ , μ ⟩ + 2 n
Lemma 2.4 ([27 Prop. 5.4.3] ). We have an isomorphism of M k ; n ( v , w ) \mathcal{M}_{k;n}(\mathbf{v},\mathbf{w}) M k ; n ( v , w ) G ( n , Q ) → M ( v − n e k , w ) ∘ G(n,Q)\to\mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})^\circ G ( n , Q ) → M ( v − n e k , w ) ∘ n n n Q Q Q M ( v − n e k , w ) × M k ; ≤ n ( v , w ) \mathcal{M}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w})\times\mathcal{M}_{k;\leq n}(\mathbf{v},\mathbf{w}) M ( v − n e k , w ) × M k ; ≤ n ( v , w )
The same result holds with M \mathcal{M} M L \mathcal{L} L L k ; n ( v , w ) \mathcal{L}_{k;n}(\mathbf{v},\mathbf{w}) L k ; n ( v , w ) L k ; 0 ( v − n e k , w ) \mathcal{L}_{k;0}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) L k ; 0 ( v − n e k , w ) [27 Prop. 5.4.3] and [27 Lem. 2.5.3] . As observed in the proof of [27 Prop. 12.3.2] , the lemma below follows from [27 Lem. 2.9.4(1)] .
Lemma 2.5. If v ≠ 0 \mathbf{v}\ne0 v = 0 x ∈ L ( v , w ) x\in\mathcal{L}(\mathbf{v},\mathbf{w}) x ∈ L ( v , w ) ϵ k ( x ) > 0 \epsilon_k(x)>0 ϵ k ( x ) > 0 k ∈ I k\in I k ∈ I
Whenever we take a Grassmannian bundle of a vector bundle, the effect on Chow groups is easily understood as a relative version of the Schubert cell description of the Chow group of the Grassmannian itself.
Let Q Q Q m m m B B B p : G ( n , Q ) → B p:G(n,Q)\to B p : G ( n , Q ) → B n n n S S S
Let Sym n , m − n \operatorname{Sym}_{n,m-n} Sym n , m − n S n × S m − n \mathfrak{S}_n\times\mathfrak{S}_{m-n} S n × S m − n ν \nu ν Δ ν ( c ) = det ( c ν a + b − a ) a , b \Delta_\nu(c)=\det(c_{\nu_a+b-a})_{a,b} Δ ν ( c ) = det ( c ν a + b − a ) a , b Sym m \operatorname{Sym}_m Sym m Sym n , m − n \operatorname{Sym}_{n,m-n} Sym n , m − n Δ ν ( c ) \Delta_\nu(c) Δ ν ( c ) c j c_j c j j j j x 1 , … , x n x_1,\ldots,x_n x 1 , … , x n ν \nu ν m − n m-n m − n n n n A G L m ∗ ( G ( n , C m ) ) ≅ Sym n , m − n A^*_{GL_m}(G(n,\mathbb{C}^m))\cong\operatorname{Sym}_{n,m-n} A G L m ∗ ( G ( n , C m )) ≅ Sym n , m − n
Lemma 2.6. The group A ∗ ( G ( n , Q ) ) A_*(G(n,Q)) A ∗ ( G ( n , Q )) f ( S ) ∩ p ∗ a f(S)\cap p^*a f ( S ) ∩ p ∗ a f ∈ Sym n f\in\operatorname{Sym}_n f ∈ Sym n a ∈ A ∗ ( B ) a\in A_*(B) a ∈ A ∗ ( B )
Proof. By Fulton’s basis theorem [13 Prop. 14.6.5] , every class in A ∗ ( G ( n , Q ) ) A_*(G(n,Q)) A ∗ ( G ( n , Q )) Δ λ ∩ p ∗ a λ \Delta_\lambda\cap p^*a_\lambda Δ λ ∩ p ∗ a λ a λ ∈ A ∗ ( B ) a_\lambda\in A_*(B) a λ ∈ A ∗ ( B ) Δ λ \Delta_\lambda Δ λ p ∗ Q / S − p ∗ Q p^*Q/S-p^*Q p ∗ Q / S − p ∗ Q c ( p ∗ Q / S − p ∗ Q ) = c ( S ) − 1 c(p^*Q/S-p^*Q)=c(S)^{-1} c ( p ∗ Q / S − p ∗ Q ) = c ( S ) − 1 S S S [1 Prop. 4.5.1] for the corresponding presentation of the cohomology of a Grassmann bundle. □ \square □
This lemma builds in our convolution mechanism:
Lemma 2.7. The map x ⊗ f ↦ F k ( n ) [ f ] ( x ) x\otimes f\mapsto F_k^{(n)}[f](x) x ⊗ f ↦ F k ( n ) [ f ] ( x ) A ∗ ( L ( v − n e k , w ) ) ⊗ Sym n → A ∗ ( L ( v , w ) ) A_*(\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}))\otimes\operatorname{Sym}_n\to A_*(\mathcal{L}(\mathbf{v},\mathbf{w})) A ∗ ( L ( v − n e k , w )) ⊗ Sym n → A ∗ ( L ( v , w )) L k ; 0 ( v − n e k , w ) \mathcal{L}_{k;0}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) L k ; 0 ( v − n e k , w ) L k ; ≤ n ( v , w ) \mathcal{L}_{k;\leq n}(\mathbf{v},\mathbf{w}) L k ; ≤ n ( v , w )
A ∗ ( L ( v − n e k , w ) ) ⊗ Sym n → F k ( n ) A ∗ ( L ( v , w ) ) res ⊗ 1 ↓ ↓ res A ∗ ( L k ; 0 ( v − n e k , w ) ) ⊗ Sym n → ψ A ∗ ( L k ; n ( v , w ) ) → ι ∗ A ∗ ( L k ; ≤ n ( v , w ) ) \begin{CD}
A_*(\mathcal{L}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}))\otimes\operatorname{Sym}_n @>{F_k^{(n)}}>> A_*(\mathcal{L}(\mathbf{v},\mathbf{w})) \\
@V{\operatorname{res}\otimes1}VV @VV{\operatorname{res}}V \\
A_*(\mathcal{L}_{k;0}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}))\otimes\operatorname{Sym}_n @>{\psi}>> A_*(\mathcal{L}_{k;n}(\mathbf{v},\mathbf{w})) @>{\iota_*}>> A_*(\mathcal{L}_{k;\leq n}(\mathbf{v},\mathbf{w}))
\end{CD} A ∗ ( L ( v − n e k , w )) ⊗ Sym n res ⊗ 1 ↓ ⏐ A ∗ ( L k ; 0 ( v − n e k , w )) ⊗ Sym n F k ( n ) ψ A ∗ ( L ( v , w )) ↓ ⏐ res A ∗ ( L k ; n ( v , w )) ι ∗ A ∗ ( L k ; ≤ n ( v , w )) where ψ ( c ⊗ f ) = f ( S ) ∩ p ∗ c \psi(c\otimes f)=f(S)\cap p^*c ψ ( c ⊗ f ) = f ( S ) ∩ p ∗ c p : L k ; n ( v , w ) → L k ; 0 ( v − n e k , w ) p:\mathcal{L}_{k;n}(\mathbf{v},\mathbf{w})\to\mathcal{L}_{k;0}(\mathbf{v}-n\mathbf{e}_k,\mathbf{w}) p : L k ; n ( v , w ) → L k ; 0 ( v − n e k , w ) S S S ι : L k ; n ( v , w ) ↪ L k ; ≤ n ( v , w ) \iota:\mathcal{L}_{k;n}(\mathbf{v},\mathbf{w})\hookrightarrow\mathcal{L}_{k;\leq n}(\mathbf{v},\mathbf{w}) ι : L k ; n ( v , w ) ↪ L k ; ≤ n ( v , w ) ι ∗ \iota_* ι ∗ res ∘ F k ( n ) \operatorname{res}\circ F_k^{(n)} res ∘ F k ( n )
Proof. Restriction to an open subset is flat pullback, which commutes with proper pushforward, refined Gysin maps, and capping by Chern classes [13 Prop. 1.7, Th. 6.2(b)] . Thus we may compute the restriction of F k ( n ) [ f ] ( c ) F_k^{(n)}[f](c) F k ( n ) [ f ] ( c ) L k ; ≤ n ( v , w ) \mathcal{L}_{k;\leq n}(\mathbf{v},\mathbf{w}) L k ; ≤ n ( v , w ) P \mathcal{P} P M ( v − n e k , w ) × M k ; ≤ n ( v , w ) \mathcal{M}(\mathbf{v}-n e_k,\mathbf{w})\times\mathcal{M}_{k;\leq n}(\mathbf{v},\mathbf{w}) M ( v − n e k , w ) × M k ; ≤ n ( v , w ) Lemma 2.4 , this restriction is the graph of the smooth morphism
g : M k ; n ( v , w ) → p M ( v − n e k , w ) ∘ ↪ M ( v − n e k , w ) , g:\mathcal{M}_{k;n}(\mathbf{v},\mathbf{w}) \xrightarrow{p} \mathcal{M}(\mathbf{v}-n e_k,\mathbf{w})^\circ\hookrightarrow\mathcal{M}(\mathbf{v}-n e_k,\mathbf{w}), g : M k ; n ( v , w ) p M ( v − n e k , w ) ∘ ↪ M ( v − n e k , w ) , and S \mathcal{S} S p p p g g g M ( v − n e k , w ) ∘ \mathcal{M}(\mathbf{v}-n e_k,\mathbf{w})^\circ M ( v − n e k , w ) ∘ M ( v − n e k , w ) \mathcal{M}(\mathbf{v}-n e_k,\mathbf{w}) M ( v − n e k , w ) M ( v − n e k , w ) ∘ \mathcal{M}(\mathbf{v}-n e_k,\mathbf{w})^\circ M ( v − n e k , w ) ∘ c c c c ∘ c^\circ c ∘ L k ; 0 ( v − n e k , w ) \mathcal{L}_{k;0}(\mathbf{v}-n e_k,\mathbf{w}) L k ; 0 ( v − n e k , w )
Refined intersection with the graph of g g g c ∘ × [ Γ g ] c^\circ\times[\Gamma_g] c ∘ × [ Γ g ] c ∘ c^\circ c ∘ g g g [13 §8.1] . Since g g g g ∗ c ∘ g^*c^\circ g ∗ c ∘ g − 1 ( L k ; 0 ) = L k ; n g^{-1}(\mathcal{L}_{k;0})=\mathcal{L}_{k;n} g − 1 ( L k ; 0 ) = L k ; n p ∗ c ∘ p^*c^\circ p ∗ c ∘ f ( S ) f(\mathcal{S}) f ( S ) q q q Definition 2.2 restricts to ι \iota ι ψ \psi ψ Lemmas 2.4 and 2.6 . The map res ⊗ 1 \operatorname{res}\otimes1 res ⊗ 1 [13 Prop. 1.8] . Since res ∘ F k ( n ) = ι ∗ ∘ ψ ∘ ( res ⊗ 1 ) \operatorname{res}\circ F_k^{(n)}=\iota_*\circ\psi\circ(\operatorname{res}\otimes1) res ∘ F k ( n ) = ι ∗ ∘ ψ ∘ ( res ⊗ 1 ) ι ∗ \iota_* ι ∗ res ∘ F k ( n ) \operatorname{res}\circ F_k^{(n)} res ∘ F k ( n )
Cyclicity of the Chow groups of the core
The following is an integral, Chow-theoretic version of [27 Prop. 12.3.2] .
Theorem 2.8. For every v \mathbf{v} v the group A ∗ ( L ( v , w ) ) A_*(\mathcal{L}(\mathbf{v},\mathbf{w})) A ∗ ( L ( v , w )) is spanned over Z \mathbb{Z} Z by the classes
F k 1 ( n 1 ) [ f 1 ] F k 2 ( n 2 ) [ f 2 ] ⋯ F k s ( n s ) [ f s ] [ L ( 0 , w ) ] , F_{k_1}^{(n_1)}[f_1]F_{k_2}^{(n_2)}[f_2]\cdots F_{k_s}^{(n_s)}[f_s][\mathcal{L}(0,\mathbf{w})], F k 1 ( n 1 ) [ f 1 ] F k 2 ( n 2 ) [ f 2 ] ⋯ F k s ( n s ) [ f s ] [ L ( 0 , w )] , where ∑ a n a e k a = v \sum_a n_a e_{k_a}=\mathbf{v} ∑ a n a e k a = v and f a ∈ Sym n a f_a\in\operatorname{Sym}_{n_a} f a ∈ Sym n a
Proof. Let A ( v ) ⊂ A ∗ ( L ( v , w ) ) A(\mathbf{v})\subset A_*(\mathcal{L}(\mathbf{v},\mathbf{w})) A ( v ) ⊂ A ∗ ( L ( v , w )) F k ( n ) [ f ] ( A ( v − n e k ) ) ⊂ A ( v ) F_k^{(n)}[f](A(\mathbf{v}-n e_k))\subset A(\mathbf{v}) F k ( n ) [ f ] ( A ( v − n e k )) ⊂ A ( v ) f ∈ Sym n f\in\operatorname{Sym}_n f ∈ Sym n A ( v ) = A ∗ ( L ( v , w ) ) A(\mathbf{v})=A_*(\mathcal{L}(\mathbf{v},\mathbf{w})) A ( v ) = A ∗ ( L ( v , w )) ∣ v ∣ = ∑ k v k |\mathbf{v}|=\sum_k v_k ∣ v ∣ = ∑ k v k v = 0 \mathbf{v}=0 v = 0 L ( 0 , w ) \mathcal{L}(0,\mathbf{w}) L ( 0 , w )
Suppose v ≠ 0 \mathbf{v}\neq0 v = 0 A ∗ ( L ( v , w ) ) A_*(\mathcal{L}(\mathbf{v},\mathbf{w})) A ∗ ( L ( v , w )) [ Y ] [Y] [ Y ] Y ⊂ L ( v , w ) Y\subset\mathcal{L}(\mathbf{v},\mathbf{w}) Y ⊂ L ( v , w ) Y Y Y N ( Y ) = max k ϵ k ( Y ) N(Y)=\max_k\epsilon_k(Y) N ( Y ) = max k ϵ k ( Y ) ϵ k ( Y ) \epsilon_k(Y) ϵ k ( Y ) ϵ k \epsilon_k ϵ k Y Y Y Lemma 2.5 , we have N ( Y ) ≥ 1 N(Y)\geq1 N ( Y ) ≥ 1 N ( Y ) ≤ max k v k N(Y)\leq\max_k v_k N ( Y ) ≤ max k v k N ( Y ) N(Y) N ( Y )
For each fixed v \mathbf{v} v [ Y ] ∈ A ( v ) [Y]\in A(\mathbf{v}) [ Y ] ∈ A ( v ) N ( Y ) N(Y) N ( Y ) n n n [ Y ′ ] ∈ A ( v ) [Y']\in A(\mathbf{v}) [ Y ′ ] ∈ A ( v ) Y ′ ⊂ L ( v , w ) Y'\subset\mathcal{L}(\mathbf{v},\mathbf{w}) Y ′ ⊂ L ( v , w ) N ( Y ′ ) > n N(Y')>n N ( Y ′ ) > n
Now consider Y Y Y N ( Y ) = n N(Y)=n N ( Y ) = n k k k ϵ k ( Y ) = n \epsilon_k(Y)=n ϵ k ( Y ) = n U = L k ; ≤ n ( v , w ) U=\mathcal{L}_{k;\leq n}(\mathbf{v},\mathbf{w}) U = L k ; ≤ n ( v , w ) Z = L k ; ≥ n + 1 ( v , w ) Z=\mathcal{L}_{k;\geq n+1}(\mathbf{v},\mathbf{w}) Z = L k ; ≥ n + 1 ( v , w ) L ( v , w ) \mathcal{L}(\mathbf{v},\mathbf{w}) L ( v , w ) Y ∘ = Y ∩ U ≠ ∅ Y^\circ=Y\cap U\neq\varnothing Y ∘ = Y ∩ U = ∅ Y Y Y L k ; n ( v , w ) \mathcal{L}_{k;n}(\mathbf{v},\mathbf{w}) L k ; n ( v , w )
By Lemma 2.4 , L k ; n ( v , w ) \mathcal{L}_{k;n}(\mathbf{v},\mathbf{w}) L k ; n ( v , w ) L k ; 0 ( v − n e k , w ) \mathcal{L}_{k;0}(\mathbf{v}-n e_k,\mathbf{w}) L k ; 0 ( v − n e k , w ) S \mathcal{S} S Lemma 2.6 ,
[ Y ∘ ] = ∑ ℓ f ℓ ( S ) ∩ p ∗ a ℓ , f ℓ ∈ Sym n , a ℓ ∈ A ∗ ( L k ; 0 ( v − n e k , w ) ) . [Y^\circ]=\sum_\ell f_\ell(\mathcal{S})\cap p^*a_\ell,\qquad f_\ell\in\operatorname{Sym}_n,\quad a_\ell\in A_*(\mathcal{L}_{k;0}(\mathbf{v}-n e_k,\mathbf{w})). [ Y ∘ ] = ℓ ∑ f ℓ ( S ) ∩ p ∗ a ℓ , f ℓ ∈ Sym n , a ℓ ∈ A ∗ ( L k ; 0 ( v − n e k , w )) . Since L k ; 0 ( v − n e k , w ) \mathcal{L}_{k;0}(\mathbf{v}-n e_k,\mathbf{w}) L k ; 0 ( v − n e k , w ) L ( v − n e k , w ) \mathcal{L}(\mathbf{v}-n e_k,\mathbf{w}) L ( v − n e k , w ) A ∗ ( L ( v − n e k , w ) ) → A ∗ ( L k ; 0 ( v − n e k , w ) ) A_*(\mathcal{L}(\mathbf{v}-n e_k,\mathbf{w}))\to A_*(\mathcal{L}_{k;0}(\mathbf{v}-n e_k,\mathbf{w})) A ∗ ( L ( v − n e k , w )) → A ∗ ( L k ; 0 ( v − n e k , w )) [13 Prop. 1.8] , so we can choose b ℓ ∈ A ∗ ( L ( v − n e k , w ) ) b_\ell\in A_*(\mathcal{L}(\mathbf{v}-n e_k,\mathbf{w})) b ℓ ∈ A ∗ ( L ( v − n e k , w )) a ℓ a_\ell a ℓ ∣ v ∣ |\mathbf{v}| ∣ v ∣ b ℓ ∈ A ( v − n e k ) b_\ell\in A(\mathbf{v}-n e_k) b ℓ ∈ A ( v − n e k )
D = ∑ ℓ F k ( n ) [ f ℓ ] ( b ℓ ) ∈ A ( v ) . D=\sum_\ell F_k^{(n)}[f_\ell](b_\ell)\in A(\mathbf{v}). D = ℓ ∑ F k ( n ) [ f ℓ ] ( b ℓ ) ∈ A ( v ) . Now, we consider the behavior of this class in the exact sequence [13 Prop. 1.8]
A ∗ ( Z ) ⟶ A ∗ ( L ( v , w ) ) ⟶ A ∗ ( U ) ⟶ 0. A_*(Z)\longrightarrow A_*(\mathcal{L}(\mathbf{v},\mathbf{w}))\longrightarrow A_*(U)\longrightarrow0. A ∗ ( Z ) ⟶ A ∗ ( L ( v , w )) ⟶ A ∗ ( U ) ⟶ 0. Applying the commutative diagram of Lemma 2.7 to each term of D D D b ℓ ∣ L k ; 0 ( v − n e k , w ) = a ℓ b_\ell|_{\mathcal{L}_{k;0}(\mathbf{v}-n e_k,\mathbf{w})}=a_\ell b ℓ ∣ L k ; 0 ( v − n e k , w ) = a ℓ
D ∣ U = ι ∗ ψ ( ∑ ℓ a ℓ ⊗ f ℓ ) = ι ∗ ( ∑ ℓ f ℓ ( S ) ∩ p ∗ a ℓ ) = ι ∗ [ Y ∘ ] = [ Y ] ∣ U , D|_U=\iota_*\psi\left(\sum_\ell a_\ell\otimes f_\ell\right)=\iota_*\left(\sum_\ell f_\ell(\mathcal{S})\cap p^*a_\ell\right)=\iota_*[Y^\circ]=[Y]|_U, D ∣ U = ι ∗ ψ ( ℓ ∑ a ℓ ⊗ f ℓ ) = ι ∗ ( ℓ ∑ f ℓ ( S ) ∩ p ∗ a ℓ ) = ι ∗ [ Y ∘ ] = [ Y ] ∣ U , where the last equality holds since Y ∘ = Y ∩ U Y^\circ=Y\cap U Y ∘ = Y ∩ U L k ; n ( v , w ) \mathcal{L}_{k;n}(\mathbf{v},\mathbf{w}) L k ; n ( v , w ) [ Y ] − D [Y]-D [ Y ] − D Z Z Z [ Y ′ ] [Y'] [ Y ′ ] Y ′ ⊂ Z Y'\subset Z Y ′ ⊂ Z Y ′ Y' Y ′ ϵ k ( Y ′ ) ≥ n + 1 \epsilon_k(Y')\geq n+1 ϵ k ( Y ′ ) ≥ n + 1 [ Y ′ ] ∈ A ( v ) [Y']\in A(\mathbf{v}) [ Y ′ ] ∈ A ( v )
This shows that as a module over these operators, the homology of the core is cyclically generated.
Corollary 2.9. For any field k \mathbf{k} k H ∗ ( L Λ ; k ) = A ∗ ( L Λ ) ⊗ Z k H_{*}(\mathcal{L}_{\Lambda};\mathbf{k})=A_{*}(\mathcal{L}_{\Lambda})\otimes_{\mathbb{Z}}\mathbf{k} H ∗ ( L Λ ; k ) = A ∗ ( L Λ ) ⊗ Z k [ L ( 0 , w ) ] [\mathcal{L}(0,\mathbf{w})] [ L ( 0 , w )] F k ( n ) [ f ] F_{k}^{(n)}[f] F k ( n ) [ f ] f ∈ Sym n f\in\operatorname{Sym}_{n} f ∈ Sym n
Since the homology of the core is dual to the cohomology of the quiver variety, we obtain a dual statement about the cohomology of the quiver variety.
Lemma 2.10. For any field k \mathbf{k} k H ∗ ( M Λ ; k ) H^{*}(\mathcal{M}_{\Lambda};\mathbf{k}) H ∗ ( M Λ ; k ) [ M ( 0 , w ) ] [\mathcal{M}(0,\mathbf{w})] [ M ( 0 , w )] H ∗ ( M Λ ; k ) H^{*}(\mathcal{M}_{\Lambda};\mathbf{k}) H ∗ ( M Λ ; k ) E k ( n ) [ f ] E_{k}^{(n)}[f] E k ( n ) [ f ] f ∈ Sym n f\in\operatorname{Sym}_{n} f ∈ Sym n
Proof. For v ∈ H ∗ ( M Λ ; k ) v\in H^{*}(\mathcal{M}_{\Lambda};\mathbf{k}) v ∈ H ∗ ( M Λ ; k ) α \alpha α v v v H ∗ ( M Λ , α ; k ) H^{*}(\mathcal{M}_{\Lambda,\alpha};\mathbf{k}) H ∗ ( M Λ , α ; k ) μ = Λ − α \mu=\Lambda-\alpha μ = Λ − α Lemma 2.1 , there is a sequence of operators F k 1 ( n 1 ) [ f 1 ] , … , F k s ( n s ) [ f s ] F_{k_{1}}^{(n_{1})}[f_{1}],\ldots,F_{k_{s}}^{(n_{s})}[f_{s}] F k 1 ( n 1 ) [ f 1 ] , … , F k s ( n s ) [ f s ]
⟨ v , F k 1 ( n 1 ) [ f 1 ] ⋯ F k s ( n s ) [ f s ] [ L ( 0 , w ) ] ⟩ ≠ 0. \left\langle v,F_{k_{1}}^{(n_{1})}[f_{1}]\cdots F_{k_{s}}^{(n_{s})}[f_{s}][\mathcal{L}(0,\mathbf{w})]\right\rangle\ne0. ⟨ v , F k 1 ( n 1 ) [ f 1 ] ⋯ F k s ( n s ) [ f s ] [ L ( 0 , w )] ⟩ = 0. By the minimality of μ \mu μ E k s ( n s ) [ f s ] ⋯ E k 1 ( n 1 ) [ f 1 ] v E_{k_{s}}^{(n_{s})}[f_{s}]\cdots E_{k_{1}}^{(n_{1})}[f_{1}]v E k s ( n s ) [ f s ] ⋯ E k 1 ( n 1 ) [ f 1 ] v H ∗ ( M ( 0 , w ) ; k ) H^{*}(\mathcal{M}(0,\mathbf{w});\mathbf{k}) H ∗ ( M ( 0 , w ) ; k ) v v v ν \nu ν ν + α \nu+\alpha ν + α H ∗ ( M Λ ; k ) H^{*}(\mathcal{M}_{\Lambda};\mathbf{k}) H ∗ ( M Λ ; k ) ν ≤ μ = Λ − α \nu\leq\mu=\Lambda-\alpha ν ≤ μ = Λ − α μ \mu μ Lemma 2.3 , the non-zero pairing
⟨ E k s ( n s ) [ f s ] ⋯ E k 1 ( n 1 ) [ f 1 ] v , [ L ( 0 , w ) ] ⟩ = ⟨ v , F k 1 ( n 1 ) [ f 1 ] ⋯ F k s ( n s ) [ f s ] [ L ( 0 , w ) ] ⟩ \left\langle E_{k_{s}}^{(n_{s})}[f_{s}]\cdots E_{k_{1}}^{(n_{1})}[f_{1}]v,[\mathcal{L}(0,\mathbf{w})]\right\rangle=\left\langle v,F_{k_{1}}^{(n_{1})}[f_{1}]\cdots F_{k_{s}}^{(n_{s})}[f_{s}][\mathcal{L}(0,\mathbf{w})]\right\rangle ⟨ E k s ( n s ) [ f s ] ⋯ E k 1 ( n 1 ) [ f 1 ] v , [ L ( 0 , w )] ⟩ = ⟨ v , F k 1 ( n 1 ) [ f 1 ] ⋯ F k s ( n s ) [ f s ] [ L ( 0 , w )] ⟩ shows that E k s ( n s ) [ f s ] ⋯ E k 1 ( n 1 ) [ f 1 ] v E_{k_{s}}^{(n_{s})}[f_{s}]\cdots E_{k_{1}}^{(n_{1})}[f_{1}]v E k s ( n s ) [ f s ] ⋯ E k 1 ( n 1 ) [ f 1 ] v H ∗ ( M ( 0 , w ) ; k ) H^{*}(\mathcal{M}(0,\mathbf{w});\mathbf{k}) H ∗ ( M ( 0 , w ) ; k ) v v v
Let x 1 , … , x n x_{1},\ldots,x_{n} x 1 , … , x n S S S f ∈ Sym n f\in\operatorname{Sym}_{n} f ∈ Sym n f = ∑ a f a x 1 a 1 ⋯ x n a n f=\sum_{\mathbf{a}}f_{\mathbf{a}}x_{1}^{a_{1}}\cdots x_{n}^{a_{n}} f = ∑ a f a x 1 a 1 ⋯ x n a n n = 1 n=1 n = 1 F k , r = F k ( 1 ) [ x 1 r ] F_{k,r}=F_{k}^{(1)}[x_{1}^{r}] F k , r = F k ( 1 ) [ x 1 r ]
Lemma 2.11. For f ∈ Sym n f\in\operatorname{Sym}_{n} f ∈ Sym n
∑ a f a F k , a 1 F k , a 2 ⋯ F k , a n = ( − 1 ) ( n 2 ) n ! F k ( n ) [ f ] \sum_{\mathbf{a}}f_{\mathbf{a}}F_{k,a_{1}}F_{k,a_{2}}\cdots F_{k,a_{n}}=(-1)^{\binom{n}{2}}n!F_{k}^{(n)}[f] a ∑ f a F k , a 1 F k , a 2 ⋯ F k , a n = ( − 1 ) ( 2 n ) n ! F k ( n ) [ f ] as operators H ∗ ( L ( v − n e k , w ) ; Z ) → H ∗ ( L ( v , w ) ; Z ) H_{*}(\mathcal{L}(\mathbf{v}-n\mathbf{e}_{k},\mathbf{w});\mathbb{Z})\to H_{*}(\mathcal{L}(\mathbf{v},\mathbf{w});\mathbb{Z}) H ∗ ( L ( v − n e k , w ) ; Z ) → H ∗ ( L ( v , w ) ; Z )
Proof. Let P ~ \widetilde{\mathcal{P}} P V 1 = U 0 ⊂ U 1 ⊂ ⋯ ⊂ U n = V 2 V^{1}=U^{0}\subset U^{1}\subset\cdots\subset U^{n}=V^{2} V 1 = U 0 ⊂ U 1 ⊂ ⋯ ⊂ U n = V 2 dim U j = v − ( n − j ) e k \dim U^{j}=\mathbf{v}-(n-j)\mathbf{e}_{k} dim U j = v − ( n − j ) e k ( V 1 , V 2 ) (V^{1},V^{2}) ( V 1 , V 2 ) P k ( n ) \mathcal{P}_{k}^{(n)} P k ( n ) V 1 V^{1} V 1 V 2 V^{2} V 2 k k k V k 1 ⊂ U k 1 ⊂ ⋯ ⊂ V k 2 V_{k}^{1}\subset U_{k}^{1}\subset\cdots\subset V_{k}^{2} V k 1 ⊂ U k 1 ⊂ ⋯ ⊂ V k 2 k k k V k 1 V_{k}^{1} V k 1 k k k V j 2 = V j 1 V_{j}^{2}=V_{j}^{1} V j 2 = V j 1 j ≠ k j\ne k j = k ρ : P ~ → P k ( n ) \rho:\widetilde{\mathcal{P}}\to\mathcal{P}_{k}^{(n)} ρ : P → P k ( n ) S = V k 2 / V k 1 S=V_{k}^{2}/V_{k}^{1} S = V k 2 / V k 1 ( n 2 ) \binom{n}{2} ( 2 n ) U k j / U k j − 1 U_{k}^{j}/U_{k}^{j-1} U k j / U k j − 1 x j x_{j} x j ρ ∗ S \rho^{*}S ρ ∗ S
Write M j = M ( v − ( n − j ) e k , w ) \mathcal{M}_{j}=\mathcal{M}(\mathbf{v}-(n-j)\mathbf{e}_{k},\mathbf{w}) M j = M ( v − ( n − j ) e k , w ) 0 ≤ j ≤ n 0\leq j\leq n 0 ≤ j ≤ n P j = P k ( 1 ) ⊂ M j − 1 × M j \mathcal{P}_{j}=\mathcal{P}_{k}^{(1)}\subset\mathcal{M}_{j-1}\times\mathcal{M}_{j} P j = P k ( 1 ) ⊂ M j − 1 × M j M \mathcal{M} M P j \mathcal{P}_{j} P j ( V j − 1 ⊂ V j ) (V^{j-1}\subset V^{j}) ( V j − 1 ⊂ V j ) P 1 × M 1 P 2 × M 2 ⋯ × M n − 1 P n \mathcal{P}_{1}\times_{\mathcal{M}_{1}}\mathcal{P}_{2}\times_{\mathcal{M}_{2}}\cdots\times_{\mathcal{M}_{n-1}}\mathcal{P}_{n} P 1 × M 1 P 2 × M 2 ⋯ × M n − 1 P n V 0 ⊂ V 1 ⊂ ⋯ ⊂ V n V^{0}\subset V^{1}\subset\cdots\subset V^{n} V 0 ⊂ V 1 ⊂ ⋯ ⊂ V n P ~ \widetilde{\mathcal{P}} P
Put P = P 1 × ⋯ × P n \mathcal{P}=\mathcal{P}_{1}\times\cdots\times\mathcal{P}_{n} P = P 1 × ⋯ × P n ∏ j = 1 n ( M j − 1 × M j ) \prod_{j=1}^{n}(\mathcal{M}_{j-1}\times\mathcal{M}_{j}) ∏ j = 1 n ( M j − 1 × M j )
δ ~ : M 0 × M 1 × ⋯ × M n ⟶ ∏ j = 1 n ( M j − 1 × M j ) , ( m 0 , … , m n ) ⟼ ( ( m 0 , m 1 ) , ( m 1 , m 2 ) , … , ( m n − 1 , m n ) ) . \widetilde{\delta}:\mathcal{M}_{0}\times\mathcal{M}_{1}\times\cdots\times\mathcal{M}_{n}\longrightarrow\prod_{j=1}^{n}(\mathcal{M}_{j-1}\times\mathcal{M}_{j}),\qquad(m_{0},\ldots,m_{n})\longmapsto((m_{0},m_{1}),(m_{1},m_{2}),\ldots,(m_{n-1},m_{n})). δ : M 0 × M 1 × ⋯ × M n ⟶ j = 1 ∏ n ( M j − 1 × M j ) , ( m 0 , … , m n ) ⟼ (( m 0 , m 1 ) , ( m 1 , m 2 ) , … , ( m n − 1 , m n )) . We can identify δ ~ = id M 0 × δ 1 × ⋯ × δ n − 1 × id M n \widetilde{\delta}=\operatorname{id}_{\mathcal{M}_{0}}\times\delta_{1}\times\cdots\times\delta_{n-1}\times\operatorname{id}_{\mathcal{M}_{n}} δ = id M 0 × δ 1 × ⋯ × δ n − 1 × id M n δ j : M j → M j × M j \delta_{j}:\mathcal{M}_{j}\to\mathcal{M}_{j}\times\mathcal{M}_{j} δ j : M j → M j × M j δ ~ \widetilde{\delta} δ ⨁ j = 1 n − 1 T M j \bigoplus_{j=1}^{n-1}T\mathcal{M}_{j} ⨁ j = 1 n − 1 T M j P ~ = δ ~ − 1 ( P ) \widetilde{\mathcal{P}}=\widetilde{\delta}^{-1}(\mathcal{P}) P = δ − 1 ( P ) δ ~ \widetilde{\delta} δ P \mathcal{P} P F k , a 1 ⋯ F k , a n F_{k,a_{1}}\cdots F_{k,a_{n}} F k , a 1 ⋯ F k , a n Definition 2.2 for the diagonal of each of M 0 , … , M n − 1 \mathcal{M}_{0},\ldots,\mathcal{M}_{n-1} M 0 , … , M n − 1 M 1 , … , M n − 1 \mathcal{M}_{1},\ldots,\mathcal{M}_{n-1} M 1 , … , M n − 1 δ j ! \delta_{j}^{!} δ j ! M 0 \mathcal{M}_{0} M 0 Definition 2.2 , with the class
γ = ρ ∗ ′ ( x n a n ⋯ x 1 a 1 ∩ δ ~ ! [ P ] ) ∈ A ∗ ( M 0 × M n ) , (5) \gamma=\rho_{*}'(x_{n}^{a_{n}}\cdots x_{1}^{a_{1}}\cap\widetilde{\delta}^{!}[\mathcal{P}])\in A_{*}(\mathcal{M}_{0}\times\mathcal{M}_{n}), \tag*{(5)} γ = ρ ∗ ′ ( x n a n ⋯ x 1 a 1 ∩ δ ! [ P ]) ∈ A ∗ ( M 0 × M n ) , (5) where ρ : P ~ → M 0 × M n \rho:\widetilde{\mathcal{P}}\to\mathcal{M}_0\times\mathcal{M}_n ρ : P → M 0 × M n x j x_j x j P P P δ ~ \widetilde{\delta} δ P ~ \widetilde{\mathcal{P}} P P P P P ~ ↪ P \widetilde{\mathcal{P}}\hookrightarrow P P ↪ P [13 Th. 6.3] gives δ ~ ! [ P ] = c t o p ( E ) ∩ [ P ~ ] \widetilde{\delta}^{!}[P]=c_{\mathrm{top}}(E)\cap[\widetilde{\mathcal{P}}] δ ! [ P ] = c top ( E ) ∩ [ P ] E E E
⨁ j = 1 n T P j ⟶ ⨁ j = 1 n − 1 T M j , ( u j ) j ⟼ ( d π 2 ( j ) u j − d π 1 ( j + 1 ) u j + 1 ) j , \bigoplus_{j=1}^{n}TP_j\longrightarrow\bigoplus_{j=1}^{n-1}T\mathcal{M}_j,\qquad(u_j)_j\longmapsto\left(d\pi^{(j)}_2u_j-d\pi^{(j+1)}_1u_{j+1}\right)_j, j = 1 ⨁ n T P j ⟶ j = 1 ⨁ n − 1 T M j , ( u j ) j ⟼ ( d π 2 ( j ) u j − d π 1 ( j + 1 ) u j + 1 ) j , restricted to P ~ \widetilde{\mathcal{P}} P π 1 ( j ) , π 2 ( j ) \pi^{(j)}_1,\pi^{(j)}_2 π 1 ( j ) , π 2 ( j ) P j P_j P j E E E dim P ~ \dim\widetilde{\mathcal{P}} dim P
We identify E E E P j P_j P j M j − 1 × M j \mathcal{M}_{j-1}\times\mathcal{M}_j M j − 1 × M j ω j − 1 ⊟ ω j \omega_{j-1}\boxminus\omega_j ω j − 1 ⊟ ω j [26 Th. 5.7] . The symplectic forms give a perfect pairing between E E E ( m 1 , … , m n − 1 ) (m_1,\ldots,m_{n-1}) ( m 1 , … , m n − 1 ) m j ∈ T M j m_j\in T\mathcal{M}_j m j ∈ T M j ( m j − 1 , m j ) (m_{j-1},m_j) ( m j − 1 , m j ) T P j TP_j T P j m 0 = m n = 0 m_0=m_n=0 m 0 = m n = 0 P j P_j P j ( m j − 1 , m j ) ∈ T P j (m_{j-1},m_j)\in TP_j ( m j − 1 , m j ) ∈ T P j P ~ → M 0 × M n \widetilde{\mathcal{P}}\to\mathcal{M}_0\times\mathcal{M}_n P → M 0 × M n P ~ \widetilde{\mathcal{P}} P M 0 × M n \mathcal{M}_0\times\mathcal{M}_n M 0 × M n P ( n ) \mathcal{P}^{(n)} P ( n ) ρ \rho ρ E ≅ T ρ ∗ E\cong T_\rho^* E ≅ T ρ ∗ c t o p ( E ) = c ( n 2 ) ( T ρ ∗ ) c_{\mathrm{top}}(E)=c_{\binom{n}{2}}(T_\rho^*) c top ( E ) = c ( 2 n ) ( T ρ ∗ ) T ρ T_\rho T ρ Hom ( U k i / U k i − 1 , U k j / U k j − 1 ) \operatorname{Hom}(U_k^i/U_k^{i-1},U_k^j/U_k^{j-1}) Hom ( U k i / U k i − 1 , U k j / U k j − 1 ) i < j i<j i < j x j − x i x_j-x_i x j − x i
c t o p ( E ) = ∏ i < j ( x i − x j ) = ( − 1 ) ( n 2 ) ∏ i < j ( x j − x i ) . c_{\mathrm{top}}(E)=\prod_{i<j}(x_i-x_j)=(-1)^{\binom{n}{2}}\prod_{i<j}(x_j-x_i). c top ( E ) = i < j ∏ ( x i − x j ) = ( − 1 ) ( 2 n ) i < j ∏ ( x j − x i ) . Integrating out the complete flag, ρ ∗ ( g ⋅ ∏ i < j ( x j − x i ) ∩ [ P ~ ] ) = { ∑ w ∈ S n w ( g ) } ∩ [ P ( n ) ] \rho_*\left(g\cdot\prod_{i<j}(x_j-x_i)\cap[\widetilde{\mathcal{P}}]\right)=\left\{\sum_{w\in S_n}w(g)\right\}\cap[\mathcal{P}^{(n)}] ρ ∗ ( g ⋅ ∏ i < j ( x j − x i ) ∩ [ P ] ) = { ∑ w ∈ S n w ( g ) } ∩ [ P ( n ) ] g g g x j x_j x j [13 §14.2] : ρ ∗ \rho_* ρ ∗ ∂ w 0 \partial_{w_0} ∂ w 0 ∏ i < j ( x j − x i ) \prod_{i<j}(x_j-x_i) ∏ i < j ( x j − x i ) T ρ T_\rho T ρ n = 2 n=2 n = 2 g = 1 g=1 g = 1 ρ ∗ ( x 2 − x 1 ) = χ ( P 1 ) = 2 \rho_*(x_2-x_1)=\chi(\mathbb{P}^1)=2 ρ ∗ ( x 2 − x 1 ) = χ ( P 1 ) = 2 f f f ∑ w w ( x 1 a 1 ⋯ x n a n ) = n ! f \sum_w w(x_1^{a_1}\cdots x_n^{a_n})=n!\,f ∑ w w ( x 1 a 1 ⋯ x n a n ) = n ! f ∑ a f a F k , a 1 ⋯ F k , a n = ( − 1 ) ( n 2 ) n ! F k ( n ) [ f ] \sum_a f_aF_{k,a_1}\cdots F_{k,a_n}=(-1)^{\binom{n}{2}}n!F_k^{(n)}[f] ∑ a f a F k , a 1 ⋯ F k , a n = ( − 1 ) ( 2 n ) n ! F k ( n ) [ f ]
Comparison with the KLR algebras
The KLR side
We will require the following objects from the world of categorified Lie algebras:
The nilHecke algebra N H n \mathrm{NH}_n NH n n n n Z \mathbb{Z} Z y 1 , … , y n , ψ 1 , … , ψ n − 1 y_1,\ldots,y_n,\psi_1,\ldots,\psi_{n-1} y 1 , … , y n , ψ 1 , … , ψ n − 1 Mat n ! ( Sym n ) \operatorname{Mat}_{n!}(\operatorname{Sym}_n) Mat n ! ( Sym n ) Sym n = Z [ y 1 , … , y n ] S n \operatorname{Sym}_n=\mathbb{Z}[y_1,\ldots,y_n]^{S_n} Sym n = Z [ y 1 , … , y n ] S n
e n = y 1 n − 1 y 2 n − 2 ⋯ y n − 1 ψ w 0 ∈ N H n . e_n=y_1^{n-1}y_2^{n-2}\cdots y_{n-1}\psi_{w_0}\in\mathrm{NH}_n. e n = y 1 n − 1 y 2 n − 2 ⋯ y n − 1 ψ w 0 ∈ NH n . The KLR algebra R α R_\alpha R α [19 , 31 ] is the Z \mathbb{Z} Z e ( i ) e(\mathbf{i}) e ( i ) i = ( i 1 , … , i n ) ∈ I n \mathbf{i}=(i_1,\ldots,i_n)\in I^n i = ( i 1 , … , i n ) ∈ I n ∑ t = 1 n α i t = α \sum_{t=1}^{n}\alpha_{i_t}=\alpha ∑ t = 1 n α i t = α y 1 , … , y n y_1,\ldots,y_n y 1 , … , y n ψ 1 , … , ψ n − 1 \psi_1,\ldots,\psi_{n-1} ψ 1 , … , ψ n − 1 Q i j ( u , v ) = ( u − v ) # j → i ( v − u ) # i → j Q_{ij}(u,v)=(u-v)^{\#{j\to i}}(v-u)^{\#{i\to j}} Q ij ( u , v ) = ( u − v ) # j → i ( v − u ) # i → j i ≠ j i\ne j i = j I I I Q i i ( u , v ) = 0 Q_{ii}(u,v)=0 Q ii ( u , v ) = 0 [34 §1.1] . Since signs can have a subtle effect on braidjunctions, we note that, for i ≠ j i\ne j i = j I I I a i j a_{ij} a ij
Q i j ( u , v ) = r i j ( u − v ) − a i j = r j i ( v − u ) − a i j , − a i j = # j → i } + # i → j } , r i j = ( − 1 ) # i → j } . Q_{ij}(u,v)=r_{ij}(u-v)^{-a_{ij}}=r_{ji}(v-u)^{-a_{ij}},\qquad-a_{ij}=\#{j\to i\}+\#{i\to j\},\qquad r_{ij}=(-1)^{\#{i\to j\}}.}}} Q ij ( u , v ) = r ij ( u − v ) − a ij = r j i ( v − u ) − a ij , − a ij = # j → i } + # i → j } , r ij = ( − 1 ) # i → j } . In particular r i j = ( − 1 ) a i j r j i r_{ij}=(-1)^{a_{ij}}r_{ji} r ij = ( − 1 ) a ij r j i [32 ] .
The cyclotomic quotient R α Λ = R α / ⟨ y 1 ⟨ α i ∨ , Λ ⟩ e ( i ) ⟩ R^\Lambda_\alpha=R_\alpha/\langle y_1^{\langle\alpha^\vee_i,\Lambda\rangle}e(\mathbf{i})\rangle R α Λ = R α / ⟨ y 1 ⟨ α i ∨ , Λ ⟩ e ( i )⟩ Λ \Lambda Λ
R Λ = ⨁ α ∈ Q + R α Λ and R n Λ = ⨁ α ∈ Q n + R α Λ , R^\Lambda=\bigoplus_{\alpha\in Q_+}R^\Lambda_\alpha
\qquad\text{and}\qquad
R^\Lambda_n=\bigoplus_{\alpha\in Q^+_n}R^\Lambda_\alpha, R Λ = α ∈ Q + ⨁ R α Λ and R n Λ = α ∈ Q n + ⨁ R α Λ , so that Z ( R Λ ) = ⨁ α Z ( R α Λ ) Z(R^\Lambda)=\bigoplus_\alpha Z(R^\Lambda_\alpha) Z ( R Λ ) = ⨁ α Z ( R α Λ ) Tr ( R Λ ) = ⨁ α Tr ( R α Λ ) \operatorname{Tr}(R^\Lambda)=\bigoplus_\alpha\operatorname{Tr}(R^\Lambda_\alpha) Tr ( R Λ ) = ⨁ α Tr ( R α Λ ) R α Λ R^\Lambda_\alpha R α Λ U \mathcal{U} U μ = Λ − α \mu=\Lambda-\alpha μ = Λ − α k \mathbb{k} k R α ( k ) R_\alpha(\mathbb{k}) R α ( k ) R α Λ ( k ) R^\Lambda_\alpha(\mathbb{k}) R α Λ ( k ) k \mathbb{k} k
The Kac–Moody 2-category U \mathcal{U} U [19 , 31 ] . We use the cyclic presentation of [4 ] , since we require a cyclic biadjunction.
The divided power 1-morphisms in U \mathcal{U} U E i ( n ) = ( E i n , e n ) \mathcal{E}_i^{(n)}=(\mathcal{E}_i^n,e_n) E i ( n ) = ( E i n , e n ) E i n ≅ ( E i ( n ) ) ⊕ n ! \mathcal{E}_i^{n}\cong(\mathcal{E}_i^{(n)})^{\oplus n!} E i n ≅ ( E i ( n ) ) ⊕ n ! n ≥ 1 n\ge1 n ≥ 1
(6) We use the usual convention of letting U \mathcal{U} U U ∗ \mathcal{U}^{*} U ∗ U \mathcal{U} U U ∗ \mathcal{U}^{*} U ∗ Tr ( R α Λ ) \operatorname{Tr}(R^{\Lambda}_{\alpha}) Tr ( R α Λ ) μ = Λ − α \mu=\Lambda-\alpha μ = Λ − α R α Λ R^{\Lambda}_{\alpha} R α Λ E i \mathcal{E}_{i} E i F i \mathcal{F}_{i} F i
Lemma 3.1. The groups R Λ R^{\Lambda} R Λ Z ( R Λ ) Z(R^{\Lambda}) Z ( R Λ ) are free abelian.
Proof. The freeness of R Λ R^{\Lambda} R Λ R α Λ R^{\Lambda}_{\alpha} R α Λ [35 Cor. 3.22] , [16 ] . The center Z ( R α Λ ) Z(R^{\Lambda}_{\alpha}) Z ( R α Λ ) R α Λ R^{\Lambda}_{\alpha} R α Λ
Note that this does not imply that dim Z ( R α Λ ( k ) ) \dim Z(R^{\Lambda}_{\alpha}(k)) dim Z ( R α Λ ( k )) k k k
The trace of a Z \mathbb{Z} Z C \mathcal{C} C
Tr ( C ) = ( ⨁ X End C ( X ) ) / span { f g − g f } . \operatorname{Tr}(\mathcal{C})=\left(\bigoplus_{X}\operatorname{End}_{\mathcal{C}}(X)\right)\big/\operatorname{span}\{fg-gf\}. Tr ( C ) = ( X ⨁ End C ( X ) ) / span { f g − g f } . where f : X → Y f:X\to Y f : X → Y g : Y → X g:Y\to X g : Y → X [ f ] [f] [ f ] f f f A A A Tr ( A ) : = Tr ( A -proj ) ≅ A / [ A , A ] \operatorname{Tr}(A):=\operatorname{Tr}(A\text{-proj})\cong A/[A,A] Tr ( A ) := Tr ( A -proj ) ≅ A / [ A , A ] Tr \operatorname{Tr} Tr [5 §§2.5–2.6] . So a 2-representation of U \mathcal{U} U C = ⨁ μ C μ \mathcal{C}=\bigoplus_{\mu}\mathcal{C}_{\mu} C = ⨁ μ C μ Tr ( C ) = ⨁ μ Tr ( C μ ) \operatorname{Tr}(\mathcal{C})=\bigoplus_{\mu}\operatorname{Tr}(\mathcal{C}_{\mu}) Tr ( C ) = ⨁ μ Tr ( C μ ) Tr ( U ∗ ) \operatorname{Tr}(\mathcal{U}^{*}) Tr ( U ∗ ) [ a ] ⋅ [ g ] = [ a ∘ u ( g ) ] [a]\cdot[g]=[a\circ u(g)] [ a ] ⋅ [ g ] = [ a ∘ u ( g )] a ∈ End ( u ) a\in\operatorname{End}(u) a ∈ End ( u )
Dually, we can consider the centers Z ( C μ ) = End ( Id C μ ) Z(\mathcal{C}_{\mu})=\operatorname{End}(\operatorname{Id}_{\mathcal{C}_{\mu}}) Z ( C μ ) = End ( Id C μ ) Z \mathbb{Z} Z C μ \mathcal{C}_{\mu} C μ U \mathcal{U} U Tr ( U ∗ ) \operatorname{Tr}(\mathcal{U}^{*}) Tr ( U ∗ ) Z ( C ) = ⨁ μ Z ( C μ ) Z(\mathcal{C})=\bigoplus_{\mu}Z(\mathcal{C}_{\mu}) Z ( C ) = ⨁ μ Z ( C μ ) [5 §9] .
In particular Tr ( U ∗ ) \operatorname{Tr}(\mathcal{U}^{*}) Tr ( U ∗ ) Tr ( R Λ ) \operatorname{Tr}(R^{\Lambda}) Tr ( R Λ ) [32 §§3.2.5–3.2.6] . By [35 Th. 3.18 and Rem. 3.19] , we have a canonical duality between cocenters and centers induced by closing up elements of the cyclotomic quotient. Under this duality, the induced action of Tr ( U ∗ ) \operatorname{Tr}(\mathcal{U}^{*}) Tr ( U ∗ ) Z ( C ) Z(\mathcal{C}) Z ( C ) Tr ( C ) \operatorname{Tr}(\mathcal{C}) Tr ( C ) Lemma 3.7 ).
The trace of U ∗ \mathcal{U}^{*} U ∗ Γ \Gamma Γ U ˙ ( g [ t ] ) \dot{U}(\mathfrak{g}[t]) U ˙ ( g [ t ]) [5 Th. B] . Outside of finite type, the relations it satisfies are those of the loop algebra of Definition 3.4 below (Proposition 3.8).
For f ∈ Sym n f\in\operatorname{Sym}_{n} f ∈ Sym n
E i ( n ) [ f ] = [ f ( y ) e n ] ∈ Tr ( U ∗ ) , E^{(n)}_{i}[f]=[f(y)e_{n}]\in\operatorname{Tr}(\mathcal{U}^{*}), E i ( n ) [ f ] = [ f ( y ) e n ] ∈ Tr ( U ∗ ) , the class of the endomorphism f ( y ) e n f(y)e_{n} f ( y ) e n E i n 1 μ \mathcal{E}_{i}^{n}1_{\mu} E i n 1 μ E i , r = E i ( 1 ) [ y r ] E_{i,r}=E_{i}^{(1)}[y^{r}] E i , r = E i ( 1 ) [ y r ] F i ( n ) [ f ] F_{i}^{(n)}[f] F i ( n ) [ f ] F i , r F_{i,r} F i , r F i n \mathcal{F}_{i}^{n} F i n U ∗ \mathcal{U}^{*} U ∗
Lemma 3.2. For f ∈ Sym n f\in\operatorname{Sym}_{n} f ∈ Sym n ∑ a f a E i , a 1 ⋯ E i , a n = n ! E i ( n ) [ f ] \sum_{a} f_{a}E_{i,a_{1}}\cdots E_{i,a_{n}}=n!E_{i}^{(n)}[f] ∑ a f a E i , a 1 ⋯ E i , a n = n ! E i ( n ) [ f ] Tr ( U ∗ ) \operatorname{Tr}(\mathcal{U}^{*}) Tr ( U ∗ )
Proof. Since the product in Tr ( U ∗ ) \operatorname{Tr}(\mathcal{U}^{*}) Tr ( U ∗ ) [ f ( y ) ] [f(y)] [ f ( y )] f ( y 1 , … , y n ) f(y_{1},\ldots,y_{n}) f ( y 1 , … , y n ) E i n 1 μ \mathcal{E}_{i}^{n}1_{\mu} E i n 1 μ N H n \mathrm{NH}_{n} NH n End Sym n ( Z [ y 1 , … , y n ] ) \operatorname{End}_{\operatorname{Sym}_{n}}(\mathbb{Z}[y_{1},\ldots,y_{n}]) End Sym n ( Z [ y 1 , … , y n ]) n ! n! n ! Sym n \operatorname{Sym}_{n} Sym n e n e_{n} e n [19 , 31 ] . Thus 1 = ∑ j = 1 n ! u j v j 1=\sum_{j=1}^{n!}u_{j}v_{j} 1 = ∑ j = 1 n ! u j v j v j u j = e n v_{j}u_{j}=e_{n} v j u j = e n j j j f f f N H n \mathrm{NH}_{n} NH n
[ f ] = ∑ j [ f u j v j ] = ∑ j [ u j v j f ] = ∑ j [ f u j v j ] = n ! [ f e n ] . [f]=\sum_{j}[fu_{j}v_{j}]=\sum_{j}[u_{j}v_{j}f]=\sum_{j}[fu_{j}v_{j}]=n![fe_{n}]. [ f ] = j ∑ [ f u j v j ] = j ∑ [ u j v j f ] = j ∑ [ f u j v j ] = n ! [ f e n ] . As in [34 §1] (see also [5 §9] ), a cyclic biadjunction on U \mathcal{U} U Z ( R Λ ) Z(R^{\Lambda}) Z ( R Λ ) Tr ( U ∗ ) \operatorname{Tr}(\mathcal{U}^{*}) Tr ( U ∗ ) [ a ] [a] [ a ] a ⋆ − a\star- a ⋆ −
Comparison of the actions
Fix a bilinear form β : Z I × Z I → Z / 2 Z \beta:\mathbb{Z}I\times\mathbb{Z}I\to\mathbb{Z}/2\mathbb{Z} β : Z I × Z I → Z /2 Z β ( γ , δ ) + β ( δ , γ ) ≡ ⟨ γ , δ ⟩ ( m o d 2 ) \beta(\gamma,\delta)+\beta(\delta,\gamma)\equiv\langle\gamma,\delta\rangle\pmod2 β ( γ , δ ) + β ( δ , γ ) ≡ ⟨ γ , δ ⟩ ( mod 2 ) [34 ] . Varagnolo’s operators [33 (4.1)] , in the normalization of [34 (3.1)] , are
x k , r + = ( − 1 ) β ( μ − Λ , α k ) ( π 1 ) ∗ ( z r ∪ π 2 ∗ ( − ) ) , x k , r − = ( − 1 ) β ( α k , μ − Λ ) ( π 2 ) ∗ ( z r ∪ π 1 ∗ ( − ) ) x^{+}_{k,r}=(-1)^{\beta(\mu-\Lambda,\alpha_{k})}(\pi_{1})_{*}(z^{r}\cup\pi_{2}^{*}(-)),\qquad x^{-}_{k,r}=(-1)^{\beta(\alpha_{k},\mu-\Lambda)}(\pi_{2})_{*}(z^{r}\cup\pi_{1}^{*}(-)) x k , r + = ( − 1 ) β ( μ − Λ , α k ) ( π 1 ) ∗ ( z r ∪ π 2 ∗ ( − )) , x k , r − = ( − 1 ) β ( α k , μ − Λ ) ( π 2 ) ∗ ( z r ∪ π 1 ∗ ( − )) on H ∗ ( M Λ , α ; C ) H^{*}(\mathcal{M}_{\Lambda,\alpha};\mathbb{C}) H ∗ ( M Λ , α ; C ) π 1 , π 2 \pi_{1},\pi_{2} π 1 , π 2 P k ( 1 ) ⊂ M Λ , α − α k × M Λ , α \mathcal{P}_{k}^{(1)} \subset\mathcal{M}_{\Lambda,\alpha-\alpha_{k}} \times\mathcal{M}_{\Lambda,\alpha} P k ( 1 ) ⊂ M Λ , α − α k × M Λ , α z z z V k ′ / V k V_{k}'/V_{k} V k ′ / V k p 1 , p 2 p_{1},p_{2} p 1 , p 2 z = c 1 ( S ) z=c_{1}(S) z = c 1 ( S ) π ∗ \pi_{*} π ∗ [34 §3.1] , defined through Poincaré duality on the smooth varieties P k ( 1 ) \mathcal{P}_{k}^{(1)} P k ( 1 ) M \mathcal{M} M (4) . Comparing with (4) , we thus have
x k , r + = ( − 1 ) β ( μ − Λ , α k ) E k , r , x k , r − = ( − 1 ) β ( α k , μ − Λ ) F k , r (6) x_{k,r}^{+}=(-1)^{\beta(\mu-\Lambda,\alpha_{k})}E_{k,r}, \qquad x_{k,r}^{-}=(-1)^{\beta(\alpha_{k},\mu-\Lambda)}F_{k,r} \tag*{(6)} x k , r + = ( − 1 ) β ( μ − Λ , α k ) E k , r , x k , r − = ( − 1 ) β ( α k , μ − Λ ) F k , r (6) on H ∗ ( M Λ , α ; C ) H^{*}(\mathcal{M}_{\Lambda,\alpha};\mathbb{C}) H ∗ ( M Λ , α ; C ) μ \mu μ V ( Λ ) V(\Lambda) V ( Λ ) Λ \Lambda Λ k ∈ I k\in I k ∈ I n ≥ 1 n\ge1 n ≥ 1
σ + ( μ , k , n ) = ( − 1 ) ( n 2 ) ∏ t = 0 n − 1 ( − 1 ) β ( μ + t α k − Λ , α k ) , σ − ( μ , k , n ) = ( − 1 ) ( n 2 ) ∏ t = 0 n − 1 ( − 1 ) β ( α k , μ − t α k − Λ ) . \sigma^{+}(\mu,k,n)=(-1)^{\binom{n}{2}}\prod_{t=0}^{n-1}(-1)^{\beta(\mu+t\alpha_{k}-\Lambda,\alpha_{k})}, \qquad\sigma^{-}(\mu,k,n)=(-1)^{\binom{n}{2}}\prod_{t=0}^{n-1}(-1)^{\beta(\alpha_{k},\mu-t\alpha_{k}-\Lambda)}. σ + ( μ , k , n ) = ( − 1 ) ( 2 n ) t = 0 ∏ n − 1 ( − 1 ) β ( μ + t α k − Λ , α k ) , σ − ( μ , k , n ) = ( − 1 ) ( 2 n ) t = 0 ∏ n − 1 ( − 1 ) β ( α k , μ − t α k − Λ ) . Apart from the factor ( − 1 ) ( n 2 ) (-1)^{\binom{n}{2}} ( − 1 ) ( 2 n ) (6) along a string of n n n μ \mu μ β ′ \beta' β ′ [34 Th. 3.4] , so that the β \beta β β ′ \beta' β ′ U ∗ U^{*} U ∗ Z ( R Λ ) Z(R^{\Lambda}) Z ( R Λ ) β ′ \beta' β ′
Lemma 3.3. For all k k k n n n f ∈ Sym n f\in\operatorname{Sym}_{n} f ∈ Sym n ϕ \phi ϕ [34 ] intertwines σ + ( μ , k , n ) E k ( n ) [ f ] \sigma^{+}(\mu,k,n)E_{k}^{(n)}[f] σ + ( μ , k , n ) E k ( n ) [ f ] E k ( n ) [ f ] ⋆ − E_{k}^{(n)}[f]\star- E k ( n ) [ f ] ⋆ − σ − ( μ , k , n ) F k ( n ) [ f ] \sigma^{-}(\mu,k,n)F_{k}^{(n)}[f] σ − ( μ , k , n ) F k ( n ) [ f ] F k ( n ) [ f ] ⋆ − F_{k}^{(n)}[f]\star- F k ( n ) [ f ] ⋆ − μ \mu μ
Proof. For n = 1 n=1 n = 1 [34 Th. 3.4] (combining Corollaries 2.8 and 3.3 there) says that ϕ \phi ϕ x k , r ± x_{k,r}^{\pm} x k , r ± E k , r ⋆ − E_{k,r}\star- E k , r ⋆ − F k , r ⋆ − F_{k,r}\star- F k , r ⋆ − (6) , x k , r ± = σ ± ( μ , k , 1 ) E k , r x_{k,r}^{\pm}=\sigma^{\pm}(\mu,k,1)E_{k,r} x k , r ± = σ ± ( μ , k , 1 ) E k , r σ ± ( μ , k , 1 ) F k , r \sigma^{\pm}(\mu,k,1)F_{k,r} σ ± ( μ , k , 1 ) F k , r n n n μ \mu μ ( − 1 ) ( n 2 ) σ ± ( μ , k , n ) (-1)^{\binom{n}{2}}\sigma^{\pm}(\mu,k,n) ( − 1 ) ( 2 n ) σ ± ( μ , k , n ) σ ± ( μ , k , n ) \sigma^{\pm}(\mu,k,n) σ ± ( μ , k , n ) ( − 1 ) ( n 2 ) (-1)^{\binom{n}{2}} ( − 1 ) ( 2 n ) ∑ a f a E k , a n ⋯ E k , a 1 = ( − 1 ) ( n 2 ) n ! E k ( n ) [ f ] \sum_{\mathbf a}f_{\mathbf a}E_{k,a_n}\cdots E_{k,a_1}=(-1)^{\binom{n}{2}}n!E_{k}^{(n)}[f] ∑ a f a E k , a n ⋯ E k , a 1 = ( − 1 ) ( 2 n ) n ! E k ( n ) [ f ] F F F Q \mathbb{Q} Q ∑ a f a x k , a n ± ⋯ x k , a 1 ± = n ! σ ± ( μ , k , n ) E k ( n ) [ f ] \sum_{\mathbf a}f_{\mathbf a}x_{k,a_n}^{\pm}\cdots x_{k,a_1}^{\pm}=n!\sigma^{\pm}(\mu,k,n)E_{k}^{(n)}[f] ∑ a f a x k , a n ± ⋯ x k , a 1 ± = n ! σ ± ( μ , k , n ) E k ( n ) [ f ] F F F Tr ( U ∗ ) \operatorname{Tr}(U^{*}) Tr ( U ∗ ) n ! n! n ! Q \mathbb{Q} Q n n n
The loop algebra
Let ( a i j ) i , j ∈ I (a_{ij})_{i,j\in I} ( a ij ) i , j ∈ I Γ \Gamma Γ − a i j -a_{ij} − a ij i ≠ j i\ne j i = j
Definition 3.4. Let L g \mathfrak{L}\mathfrak{g} Lg Q \mathbb{Q} Q x i , r ± x_{i,r}^{\pm} x i , r ± ξ i , r \xi_{i,r} ξ i , r i ∈ I i\in I i ∈ I r ≥ 0 r\ge0 r ≥ 0 i , j ∈ I i,j\in I i , j ∈ I r , s ≥ 0 r,s\ge0 r , s ≥ 0
(L1)
[ ξ i , r , ξ j , s ] = 0 , [\xi_{i,r},\xi_{j,s}]=0, [ ξ i , r , ξ j , s ] = 0 , (L2)
[ ξ i , r , x j , s ± ] = ± a i j x j , r + s ± , [\xi_{i,r},x_{j,s}^{\pm}]=\pm a_{ij}x_{j,r+s}^{\pm}, [ ξ i , r , x j , s ± ] = ± a ij x j , r + s ± , (L3)
[ x i , r + , x j , s − ] = δ i j ξ i , r + s , [x_{i,r}^{+},x_{j,s}^{-}]=\delta_{ij}\xi_{i,r+s}, [ x i , r + , x j , s − ] = δ ij ξ i , r + s , (L4)
[ x i , r ± , x i , s ± ] = 0 , [x_{i,r}^{\pm},x_{i,s}^{\pm}]=0, [ x i , r ± , x i , s ± ] = 0 , (L5)
∑ p = 0 m ( − 1 ) p ( m p ) [ x i , r + m − p ± , x j , s + p ± ] = 0 for i ≠ j , m = − a i j , \sum_{p=0}^{m}(-1)^{p}\binom{m}{p}[x_{i,r+m-p}^{\pm},x_{j,s+p}^{\pm}]=0 \qquad\text{for }i\ne j,\ m=-a_{ij}, p = 0 ∑ m ( − 1 ) p ( p m ) [ x i , r + m − p ± , x j , s + p ± ] = 0 for i = j , m = − a ij , (L6)
[ x i , r 1 ± , [ x i , r 2 ± , … , [ x i , r m ± , x j , s ± ] … ] ] = 0 for i ≠ j , m = 1 − a i j , r 1 , … , r m ≥ 0. [x_{i,r_{1}}^{\pm},[x_{i,r_{2}}^{\pm},\ldots,[x_{i,r_{m}}^{\pm},x_{j,s}^{\pm}]\ldots]]=0 \qquad\text{for }i\ne j,\ m=1-a_{ij},\ r_{1},\ldots,r_{m}\ge0. [ x i , r 1 ± , [ x i , r 2 ± , … , [ x i , r m ± , x j , s ± ] … ]] = 0 for i = j , m = 1 − a ij , r 1 , … , r m ≥ 0. In terms of the series x i ± ( z ) = ∑ r ≥ 0 x i , r ± z − r − 1 x_{i}^{\pm}(z)=\sum_{r\ge0}x_{i,r}^{\pm}z^{-r-1} x i ± ( z ) = ∑ r ≥ 0 x i , r ± z − r − 1 z − r − 1 w − s − 1 z^{-r-1}w^{-s-1} z − r − 1 w − s − 1 ( z − w ) − a i j [ x i ± ( z ) , x j ± ( w ) ] (z-w)^{-a_{ij}}[x_{i}^{\pm}(z),x_{j}^{\pm}(w)] ( z − w ) − a ij [ x i ± ( z ) , x j ± ( w )] r , s ≥ 0 r,s\ge0 r , s ≥ 0 ad x i , r ± \operatorname{ad}x_{i,r}^{\pm} ad x i , r ± r 1 , … , r m r_{1},\ldots,r_{m} r 1 , … , r m
Remark 3.5. This is the Lie algebra L g \mathfrak{L}\mathfrak{g} Lg [32 Def. 3.24] . If Γ \Gamma Γ g [ t ] \mathfrak{g}[t] g [ t ] [5 Th. 3.1] , but this fails outside of finite type. For the cyclic quiver with e ≥ 4 e\ge4 e ≥ 4 U ( L g ) U(\mathfrak{L}\mathfrak{g}) U ( Lg ) λ = β = 0 \lambda=\beta=0 λ = β = 0 [14 Def. 2.3, §2] , which is the enveloping algebra of the universal central extension of s l e [ u ± 1 , v ] \mathfrak{sl}_{e}[u^{\pm1},v] sl e [ u ± 1 , v ]
For n ≥ 1 n\ge1 n ≥ 1 f = ∑ a f a x 1 a 1 ⋯ x n a n ∈ Sym n f=\sum_{\mathbf a}f_{\mathbf a}x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\in\operatorname{Sym}_{n} f = ∑ a f a x 1 a 1 ⋯ x n a n ∈ Sym n
x i ± ( n ) [ f ] = 1 n ! ∑ a f a x i , a 1 ± ⋯ x i , a n ± ∈ U ( L g ) . x_{i}^{\pm(n)}[f]=\frac{1}{n!}\sum_{\mathbf a}f_{\mathbf a}x_{i,a_{1}}^{\pm}\cdots x_{i,a_{n}}^{\pm}\in U(\mathfrak{L}\mathfrak{g}). x i ± ( n ) [ f ] = n ! 1 a ∑ f a x i , a 1 ± ⋯ x i , a n ± ∈ U ( Lg ) . By relation (L4), if f f f k r k_r k r r r r r r r x i ± ( n ) [ f ] = ∏ r ( x i , r ± ) k r / k r ! x_i^{\pm(n)}[f] = \prod_r (x_{i,r}^{\pm})^{k_r}/k_r! x i ± ( n ) [ f ] = ∏ r ( x i , r ± ) k r / k r !
Definition 3.6. For each n ≥ 1 n \ge1 n ≥ 1 i ∈ I i \in I i ∈ I X n , i ± X_{n,i}^{\pm} X n , i ± Sym n \operatorname{Sym}_n Sym n f ∈ Sym n f \in\operatorname{Sym}_n f ∈ Sym n x i ± ( n ) [ f ] x_i^{\pm(n)}[f] x i ± ( n ) [ f ] F \mathcal{F} F ⨁ n , i X n , i ± \bigoplus_{n,i} X_{n,i}^{\pm} ⨁ n , i X n , i ± F k = F ⊗ k \mathcal{F}_k = \mathcal{F} \otimes\mathbf{k} F k = F ⊗ k
Let U Z ⊂ U ( L g ) U_{\mathbb{Z}} \subset U(L\mathfrak{g}) U Z ⊂ U ( L g ) F → U ( L g ) \mathcal{F} \to U(L\mathfrak{g}) F → U ( L g ) x i ± ( n ) [ f ] x_i^{\pm(n)}[f] x i ± ( n ) [ f ] x i ± ( n ) [ f ] x_i^{\pm(n)}[f] x i ± ( n ) [ f ] k \mathbf{k} k U k = U Z ⊗ k U_{\mathbf{k}} = U_{\mathbb{Z}} \otimes\mathbf{k} U k = U Z ⊗ k
Note that U Q = U Z ⊗ Q U_{\mathbb{Q}} = U_{\mathbb{Z}} \otimes\mathbb{Q} U Q = U Z ⊗ Q U ( L g ) U(L\mathfrak{g}) U ( L g ) U Z ⊗ Q → U ( L g ) U_{\mathbb{Z}} \otimes\mathbb{Q} \to U(L\mathfrak{g}) U Z ⊗ Q → U ( L g ) U Z U_{\mathbb{Z}} U Z Q \mathbb{Q} Q U ( L g ) U(L\mathfrak{g}) U ( L g ) x i , r ± = x i ± ( 1 ) [ x 1 r ] x_{i,r}^{\pm} = x_i^{\pm(1)}[x_1^r] x i , r ± = x i ± ( 1 ) [ x 1 r ] U ( L g ) U(L\mathfrak{g}) U ( L g ) U Z U_{\mathbb{Z}} U Z F \mathcal{F} F U ( L g ) U(L\mathfrak{g}) U ( L g ) M ⊗ Q M \otimes\mathbb{Q} M ⊗ Q M M M F \mathcal{F} F M M M U Z U_{\mathbb{Z}} U Z U k U_{\mathbf{k}} U k M ⊗ k M \otimes\mathbf{k} M ⊗ k
The ring F k \mathcal{F}_{\mathbf{k}} F k k \mathbf{k} k
on H ∗ ( M Λ ; k ) H^*(\mathcal{M}_{\Lambda};\mathbf{k}) H ∗ ( M Λ ; k ) x k + ( n ) [ f ] x_k^{+(n)}[f] x k + ( n ) [ f ] σ + ( μ , k , n ) E k ( n ) [ f ] \sigma^+(\mu,k,n)E_k^{(n)}[f] σ + ( μ , k , n ) E k ( n ) [ f ] x k − ( n ) [ f ] x_k^{-(n)}[f] x k − ( n ) [ f ] σ − ( μ , k , n ) F k ( n ) [ f ] \sigma^-(\mu,k,n)F_k^{(n)}[f] σ − ( μ , k , n ) F k ( n ) [ f ] μ \mu μ
on Z ( R Λ ( k ) ) Z(R^\Lambda(\mathbf{k})) Z ( R Λ ( k )) x k + ( n ) [ f ] x_k^{+(n)}[f] x k + ( n ) [ f ] x k − ( n ) [ f ] x_k^{-(n)}[f] x k − ( n ) [ f ] E k ( n ) [ f ] ⋆ − E_k^{(n)}[f]\star- E k ( n ) [ f ] ⋆ − F k ( n ) [ f ] ⋆ − F_k^{(n)}[f]\star- F k ( n ) [ f ] ⋆ − β ′ \beta' β ′
on Tr ( R Λ ( k ) ) \operatorname{Tr}(R^\Lambda(\mathbf{k})) Tr ( R Λ ( k )) [ g ] ↦ [ f ( y ) e n ∘ E k ( n ) ( g ) ] [g] \mapsto[f(y)e_n \circ E_k^{(n)}(g)] [ g ] ↦ [ f ( y ) e n ∘ E k ( n ) ( g )] [ g ] ↦ [ f ( y ) e n ∘ F k ( n ) ( g ) ] [g] \mapsto[f(y)e_n \circ F_k^{(n)}(g)] [ g ] ↦ [ f ( y ) e n ∘ F k ( n ) ( g )] Tr ( U ∗ ) \operatorname{Tr}(\mathcal{U}^*) Tr ( U ∗ ) [5 §2.6] .
Fix the β ′ \beta' β ′ v : ν → μ v:\nu\to\mu v : ν → μ U \mathcal{U} U Q Q Q R α ′ Λ ( k ) R_{\alpha'}^\Lambda(\mathbf{k}) R α ′ Λ ( k ) ν = Λ − α ′ \nu=\Lambda-\alpha' ν = Λ − α ′
ptr v : End ( v ( Q ) ) → End ( Q ) , h ↦ ϵ v ′ ∘ v ∗ ( h ) ∘ ι v . \operatorname{ptr}_v:\operatorname{End}(v(Q))\to\operatorname{End}(Q),\qquad h\mapsto\epsilon'_v\circ v^*(h)\circ\iota_v. ptr v : End ( v ( Q )) → End ( Q ) , h ↦ ϵ v ′ ∘ v ∗ ( h ) ∘ ι v . where ι v : id ν → v ∗ v \iota_v:\operatorname{id}_\nu\to v^*v ι v : id ν → v ∗ v ϵ v ′ : v ∗ v → id ν \epsilon'_v:v^*v\to\operatorname{id}_\nu ϵ v ′ : v ∗ v → id ν ptr v \operatorname{ptr}_v ptr v v v v Q Q Q
Figure 1.
Here the module Q Q Q Q Q Q Q Q Q ptr v ∘ ptr w = ptr w v \operatorname{ptr}_v\circ\operatorname{ptr}_w=\operatorname{ptr}_{wv} ptr v ∘ ptr w = ptr w v μ : = Λ − α \mu:=\Lambda-\alpha μ := Λ − α
t μ : R α Λ ( k ) → k , t μ ( h ) = ∑ i ptr F i e ( i ) ( e ( i ) h e ( i ) ) ∈ End ( k ) = k . t_\mu:R_\alpha^\Lambda(\mathbf{k})\to\mathbf{k},\qquad t_\mu(h)=\sum_i\operatorname{ptr}_{F_i e(i)}(e(i)h e(i))\in\operatorname{End}(\mathbf{k})=\mathbf{k}. t μ : R α Λ ( k ) → k , t μ ( h ) = i ∑ ptr F i e ( i ) ( e ( i ) h e ( i )) ∈ End ( k ) = k . Let ⟨ z , [ g ] ⟩ μ = t μ ( z g ) \langle z,[g]\rangle_\mu=t_\mu(zg) ⟨ z , [ g ] ⟩ μ = t μ ( z g ) z ∈ Z ( R α Λ ( k ) ) z\in Z(R_\alpha^\Lambda(\mathbf{k})) z ∈ Z ( R α Λ ( k )) [ g ] ∈ Tr ( R α Λ ( k ) ) [g]\in\operatorname{Tr}(R_\alpha^\Lambda(\mathbf{k})) [ g ] ∈ Tr ( R α Λ ( k ))
Lemma 3.7.
(1) The form t μ t_\mu t μ R α Λ ( k ) R_\alpha^\Lambda(\mathbf{k}) R α Λ ( k ) ⟨ − , − ⟩ μ \langle-,-\rangle_\mu ⟨ − , − ⟩ μ Z ( R α Λ ( k ) ) Z(R_\alpha^\Lambda(\mathbf{k})) Z ( R α Λ ( k )) Tr ( R α Λ ( k ) ) \operatorname{Tr}(R_\alpha^\Lambda(\mathbf{k})) Tr ( R α Λ ( k ))
(2) For u : μ → ν u:\mu\to\nu u : μ → ν a ∈ End ( u ) a\in\operatorname{End}(u) a ∈ End ( u ) z ∈ Z ( R α Λ ( k ) ) z\in Z(R_\alpha^\Lambda(\mathbf{k})) z ∈ Z ( R α Λ ( k )) f ∈ Tr ( R α ′ Λ ( k ) ) f\in\operatorname{Tr}(R_{\alpha'}^\Lambda(\mathbf{k})) f ∈ Tr ( R α ′ Λ ( k )) ν = Λ − α ′ \nu=\Lambda-\alpha' ν = Λ − α ′ ⟨ a ⋆ z , f ⟩ ν = ⟨ z , [ a ∗ ] ⋆ f ⟩ μ \langle a\star z,f\rangle_\nu=\langle z,[a^*]\star f\rangle_\mu ⟨ a ⋆ z , f ⟩ ν = ⟨ z , [ a ∗ ] ⋆ f ⟩ μ
Proof. (1) The form t μ t_\mu t μ [35 Th. 3.18] only by the signs of the twisted duality, which are constant on each summand e ( i ) R α Λ e ( j ) e(i)R^\Lambda_\alpha e(j) e ( i ) R α Λ e ( j ) β ′ \beta' β ′ [35 Rem. 3.19] .
For (2), write f = [ g ] f=[g] f = [ g ] g ∈ End ( Q ) g\in\operatorname{End}(Q) g ∈ End ( Q ) Q Q Q ν \nu ν t ν ( ( a ⋆ z ) g ) t_\nu((a\star z)g) t ν (( a ⋆ z ) g )
Figure 2.
Here the coupon z z z Z ( R α Λ ( k ) ) Z(R^\Lambda_\alpha(\mathbb{k})) Z ( R α Λ ( k )) u ⋆ Q u^\star Q u ⋆ Q a ⋆ a^\star a ⋆ a ⋆ a^\star a ⋆ u ⋆ u^\star u ⋆ a a a u u u a ⋆ z a\star z a ⋆ z Q Q Q ⟨ z , [ a ⋆ ] ⋅ f ⟩ μ \langle z,[a^\star]\cdot f\rangle_\mu ⟨ z , [ a ⋆ ] ⋅ f ⟩ μ
Proposition 3.8. (1) For every field k \mathbb{k} k E i , r E_{i,r} E i , r F i , r F_{i,r} F i , r x i , r + x^+_{i,r} x i , r + x i , r − x^-_{i,r} x i , r − ( L 1 ) (L1) ( L 1 ) ( L 6 ) (L6) ( L 6 ) k \mathbb{k} k Tr ( R Λ ( k ) ) \operatorname{Tr}(R^\Lambda(\mathbb{k})) Tr ( R Λ ( k )) Z ( R Λ ( k ) ) Z(R^\Lambda(\mathbb{k})) Z ( R Λ ( k ))
(2) The action of F F F Z ( R Λ ) Z(R^\Lambda) Z ( R Λ ) U Z U_{\mathbb{Z}} U Z U k U_{\mathbb{k}} U k Z ( R Λ ) ⊗ k Z(R^\Lambda)\otimes\mathbb{k} Z ( R Λ ) ⊗ k k \mathbb{k} k
Proof. (1) For Tr ( R Λ ( k ) ) \operatorname{Tr}(R^\Lambda(\mathbb{k})) Tr ( R Λ ( k )) [32 Th. 3.25] .
For the center, the existence of this action is noted in [32 Rem. 3.41] , but with slightly different conventions, and an implicit assumption of finite type and characteristic 0. More explicitly, since the β ′ \beta' β ′ [34 ] , [ a ] ↦ [ a ⋆ ] [a]\mapsto[a^\star] [ a ] ↦ [ a ⋆ ] Tr ( U ⋆ ) \operatorname{Tr}(U^\star) Tr ( U ⋆ ) E i , r E_{i,r} E i , r F i , r F_{i,r} F i , r Lemma 3.7 , [ a ] ⋆ − [a]\star- [ a ] ⋆ − [ a ⋆ ] [a^\star] [ a ⋆ ] x i , r + ↦ x i , r − x^+_{i,r}\mapsto x^-_{i,r} x i , r + ↦ x i , r − ξ i , r ↦ ξ i , r \xi_{i,r}\mapsto\xi_{i,r} ξ i , r ↦ ξ i , r L g L\mathfrak{g} L g
(2) By (1) with k = Q \mathbb{k}=\mathbb{Q} k = Q U ( L g ) U(L\mathfrak{g}) U ( L g ) Z ( R Λ ( Q ) ) = Z ( R Λ ) ⊗ Q Z(R^\Lambda(\mathbb{Q}))=Z(R^\Lambda)\otimes\mathbb{Q} Z ( R Λ ( Q )) = Z ( R Λ ) ⊗ Q Lemma 3.2 , applied to E E E F F F Tr ( U ⋆ ) \operatorname{Tr}(U^\star) Tr ( U ⋆ ) x k ± ( n ) [ f ] x_k^{\pm(n)}[f] x k ± ( n ) [ f ] E k ( n ) [ f ] ⋆ − E_k^{(n)}[f]\star- E k ( n ) [ f ] ⋆ − F k ( n ) [ f ] ⋆ − F_k^{(n)}[f]\star- F k ( n ) [ f ] ⋆ − Z ( R α Λ ) Z(R^\Lambda_\alpha) Z ( R α Λ ) Z ( R α Λ ) Z(R^\Lambda_\alpha) Z ( R α Λ ) R α Λ R^\Lambda_\alpha R α Λ Lemma 3.1 . Thus, we obtain an action of U Z U_{\mathbb{Z}} U Z U k U_{\mathbb{k}} U k
The integral map and injectivity
Let S = H G v ∗ ( p t ; Z ) = ⨂ i Z [ c i ; 1 , … , c i ; v i ] S=H^*_{G_v}(\mathrm{pt};\mathbb{Z})=\bigotimes_i\mathbb{Z}[c_{i;1},\ldots,c_{i;v_i}] S = H G v ∗ ( pt ; Z ) = ⨂ i Z [ c i ; 1 , … , c i ; v i ] c i ; j c_{i;j} c i ; j j j j B G v BG_v B G v κ g : S → H ∗ ( M ( v , w ) ; Z ) \kappa_g:S\to H^*(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{Z}) κ g : S → H ∗ ( M ( v , w ) ; Z ) c i ; j c_{i;j} c i ; j j j j V i V_i V i κ a : S → Z ( R α Λ ) \kappa_a:S\to Z(R^\Lambda_\alpha) κ a : S → Z ( R α Λ ) c i ; j c_{i;j} c i ; j e i ; j e_{i;j} e i ; j κ g , C : S ⊗ Z C → H ∗ ( M ( v , w ) ; C ) \kappa_{g,\mathbb{C}}:S\otimes_{\mathbb{Z}}\mathbb{C}\to H^*(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{C}) κ g , C : S ⊗ Z C → H ∗ ( M ( v , w ) ; C ) κ g \kappa_g κ g H ∗ ( M ( v , w ) ; Z ) ⊗ C ≅ H ∗ ( M ( v , w ) ; C ) H^*(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{Z})\otimes\mathbb{C}\cong H^*(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{C}) H ∗ ( M ( v , w ) ; Z ) ⊗ C ≅ H ∗ ( M ( v , w ) ; C ) κ a , C : S ⊗ C → Z ( R α Λ ( C ) ) \kappa_{a,\mathbb{C}}:S\otimes\mathbb{C}\to Z(R^\Lambda_\alpha(\mathbb{C})) κ a , C : S ⊗ C → Z ( R α Λ ( C )) κ a \kappa_a κ a Z ( R α Λ ) ⊗ C ≅ Z ( R α Λ ( C ) ) Z(R^\Lambda_\alpha)\otimes\mathbb{C}\cong Z(R^\Lambda_\alpha(\mathbb{C})) Z ( R α Λ ) ⊗ C ≅ Z ( R α Λ ( C ))
Theorem 3.9 ([24 Th. 1.2] ). The map κ g \kappa_g κ g
Lemma 3.10. There is a unique ring homomorphism ϕ Z : H ∗ ( M Λ , α ; Z ) → Z ( R α Λ ) \phi_Z:H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z})\to Z(R^\Lambda_\alpha) ϕ Z : H ∗ ( M Λ , α ; Z ) → Z ( R α Λ ) ϕ Z ( c i ; j ) = e i ; j \phi_Z(c_{i;j})=e_{i;j} ϕ Z ( c i ; j ) = e i ; j ϕ Z ⊗ C \phi_Z\otimes\mathbb{C} ϕ Z ⊗ C ϕ \phi ϕ [34 ] . It is a homomorphism of F F F Definition 3.6 .
Proof. By Theorem 3.9, to construct a homomorphism, it suffices to check that the kernel of κ g \kappa_g κ g κ a \kappa_a κ a [34 Prop. 3.5] , there is a homomorphism ϕ C : H ∗ ( M Λ ; C ) → Z ( R Λ ( C ) ) \phi_{\mathbb{C}}:H^*(\mathcal{M}_\Lambda;\mathbb{C})\to Z(R^\Lambda(\mathbb{C})) ϕ C : H ∗ ( M Λ ; C ) → Z ( R Λ ( C ))
ϕ C ∘ κ g , C = κ a , C . \phi_{\mathbb{C}}\circ\kappa_{g,\mathbb{C}}=\kappa_{a,\mathbb{C}}. ϕ C ∘ κ g , C = κ a , C . By Lemma 2.1 and Lemma 3.1 , the maps from H ∗ ( M Λ ; Z ) H^*(\mathcal{M}_\Lambda;\mathbb{Z}) H ∗ ( M Λ ; Z ) R Λ R^\Lambda R Λ
ker κ g = S ∩ ker κ g , C ⊂ S ∩ ker κ a , C = ker κ a . \ker\kappa_g=S\cap\ker\kappa_{g,\mathbb{C}}\subset S\cap\ker\kappa_{a,\mathbb{C}}=\ker\kappa_a. ker κ g = S ∩ ker κ g , C ⊂ S ∩ ker κ a , C = ker κ a . Thus ϕ Z \phi_{\mathbb{Z}} ϕ Z ϕ C \phi_{\mathbb{C}} ϕ C ϕ \phi ϕ [34 Th. 3.4] . Finally, the operators by which the generators of F \mathcal{F} F ϕ \phi ϕ Lemma 3.3 . □
For a field k k k ϕ k \phi_k ϕ k
H ∗ ( M Λ , α ; k ) = H ∗ ( M Λ , α ; Z ) ⊗ k → ϕ Z ⊗ k Z ( R α Λ ) ⊗ k → Z ( R α Λ ( k ) ) , H^*(\mathcal{M}_{\Lambda,\alpha};k)=H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z})\otimes k\xrightarrow{\phi_{\mathbb{Z}}\otimes k}Z(R^\Lambda_\alpha)\otimes k\to Z(R^\Lambda_\alpha(k)), H ∗ ( M Λ , α ; k ) = H ∗ ( M Λ , α ; Z ) ⊗ k ϕ Z ⊗ k Z ( R α Λ ) ⊗ k → Z ( R α Λ ( k )) , where the last map is induced by R α Λ ⊗ k = R α Λ ( k ) R^\Lambda_\alpha\otimes k=R^\Lambda_\alpha(k) R α Λ ⊗ k = R α Λ ( k ) ϕ k ( c i ; j ) = e i ; j \phi_k(c_{i;j})=e_{i;j} ϕ k ( c i ; j ) = e i ; j F k \mathcal{F}_k F k ϕ Z \phi_{\mathbb{Z}} ϕ Z F \mathcal{F} F
Proof of Theorem A(1)–(2). Existence, uniqueness, and the comparison with ϕ \phi ϕ ϕ k \phi_k ϕ k
By Lemma 3.10, ϕ k \phi_k ϕ k F k \mathcal{F}_k F k H ∗ ( M Λ ; k ) H^*(\mathcal{M}_\Lambda;k) H ∗ ( M Λ ; k ) Z ( R Λ ( k ) ) Z(R^\Lambda(k)) Z ( R Λ ( k )) F k \mathcal{F}_k F k H ∗ ( M Λ , 0 ; k ) H^*(\mathcal{M}_{\Lambda,0};k) H ∗ ( M Λ , 0 ; k ) ϕ k \phi_k ϕ k 1 1 1 1 1 1
By Theorem 3.9, the classes c i ; j c_{i;j} c i ; j H ∗ ( M Λ , α ; k ) H^*(\mathcal{M}_{\Lambda,\alpha};k) H ∗ ( M Λ , α ; k ) ϕ k \phi_k ϕ k ϕ k \phi_k ϕ k
dim H ∗ ( M Λ , α ; k ) ≤ dim Z ( R α Λ ( k ) ) . \dim H^*(\mathcal{M}_{\Lambda,\alpha};k)\leq\dim Z(R^\Lambda_\alpha(k)). dim H ∗ ( M Λ , α ; k ) ≤ dim Z ( R α Λ ( k )) . It is an isomorphism if and only if equality holds, which is equivalent to the reverse inequality in (2); in that case the full center coincides with the JM center. This proves (2).
Note that this establishes that the action of F k \mathcal{F}_k F k H ∗ ( M Λ ; k ) H^*(\mathcal{M}_\Lambda;k) H ∗ ( M Λ ; k ) U k U_k U k ϕ Z ⊗ k : H ∗ ( M Λ ; k ) → Z ( R Λ ) ⊗ k \phi_{\mathbb{Z}}\otimes k:H^*(\mathcal{M}_\Lambda;k)\to Z(R^\Lambda)\otimes k ϕ Z ⊗ k : H ∗ ( M Λ ; k ) → Z ( R Λ ) ⊗ k F k \mathcal{F}_k F k ϕ k \phi_k ϕ k Z ( R Λ ) ⊗ k Z(R^\Lambda)\otimes k Z ( R Λ ) ⊗ k U k U_k U k
Corollary 3.11. Let Γ \Gamma Γ g \mathfrak{g} g R α Λ ( Z ) R^\Lambda_\alpha(\mathbb{Z}) R α Λ ( Z ) Z \mathbb{Z} Z g \mathfrak{g} g R α Λ ( Z ) R^\Lambda_\alpha(\mathbb{Z}) R α Λ ( Z ) Z \mathbb{Z} Z R α Λ ( Z ) R^\Lambda_\alpha(\mathbb{Z}) R α Λ ( Z )
Proof. This follows from Theorem A(1) and (1) . □
The injectivity in Theorem A(1) also allows us to pass from a characteristic-zero isomorphism to an integral isomorphism.
Proposition 3.12. Let Γ \Gamma Γ Λ \Lambda Λ α ∈ Q + \alpha\in Q^+ α ∈ Q + ϕ k \phi_k ϕ k k k k
ϕ Z : H ∗ ( M Λ , α ; Z ) ⟶ Z ( R α Λ ( Z ) ) \phi_{\mathbb{Z}}:H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z})\longrightarrow Z(R^\Lambda_\alpha(\mathbb{Z})) ϕ Z : H ∗ ( M Λ , α ; Z ) ⟶ Z ( R α Λ ( Z )) is an isomorphism of rings and Z ( R α Λ ( Z ) ) = Z ( R α Λ ( Z ) ) J M Z(R^\Lambda_\alpha(\mathbb{Z}))=Z(R^\Lambda_\alpha(\mathbb{Z}))^{\mathrm{JM}} Z ( R α Λ ( Z )) = Z ( R α Λ ( Z ) ) JM
Proof. By Lemma 3.10, ϕ Z \phi_{\mathbb{Z}} ϕ Z Lemma 2.1 (3) and the base-change compatibility of the construction in Lemma 3.10, this identifies ϕ k \phi_k ϕ k ϕ Q ⊗ Q k \phi_{\mathbb{Q}}\otimes_{\mathbb{Q}} k ϕ Q ⊗ Q k k k k Q \mathbb{Q} Q ϕ Q \phi_{\mathbb{Q}} ϕ Q
Let z ∈ Z ( R α Λ ( Z ) ) z\in Z(R^\Lambda_\alpha(\mathbb{Z})) z ∈ Z ( R α Λ ( Z )) ϕ Q \phi_{\mathbb{Q}} ϕ Q d d d x ∈ H ∗ ( M Λ , α ; Z ) x\in H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z}) x ∈ H ∗ ( M Λ , α ; Z )
d z = ϕ Z ( x ) . (7) dz=\phi_{\mathbb{Z}}(x). \tag*{(7)} d z = ϕ Z ( x ) . (7) This equality holds over Z \mathbb{Z} Z R α Λ ( Z ) R^\Lambda_\alpha(\mathbb{Z}) R α Λ ( Z ) Lemma 3.1 and hence embeds into R α Λ ( Q ) R^\Lambda_\alpha(\mathbb{Q}) R α Λ ( Q ) d d d (7) for some such x x x x x x
Suppose that d > 1 d>1 d > 1 p p p d d d (7) modulo p p p ϕ F p ( x ˉ ) = 0 \phi_{\mathbb{F}_p}(\bar{x})=0 ϕ F p ( x ˉ ) = 0 x ˉ \bar{x} x ˉ x x x p p p x ˉ = 0 \bar{x}=0 x ˉ = 0 Lemma 2.1 (3), reduction modulo p p p H ∗ ( M Λ , α ; F p ) H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{F}_p) H ∗ ( M Λ , α ; F p )
H ∗ ( M Λ , α ; Z ) / p H ∗ ( M Λ , α ; Z ) . H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z})/pH^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z}). H ∗ ( M Λ , α ; Z ) / p H ∗ ( M Λ , α ; Z ) . Thus x = p x ′ x=px' x = p x ′ x ′ ∈ H ∗ ( M Λ , α ; Z ) x'\in H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z}) x ′ ∈ H ∗ ( M Λ , α ; Z ) (7) gives
p ( d p z − ϕ Z ( x ′ ) ) = 0. p\left(\frac{d}{p}z-\phi_{\mathbb{Z}}(x')\right)=0. p ( p d z − ϕ Z ( x ′ ) ) = 0. By Lemma 3.1 , R α Z ( Z ) R_{\alpha}^{\mathbb{Z}}(\mathbb{Z}) R α Z ( Z )
d p z = ϕ Z ( x ′ ) . \frac{d}{p}z=\phi_Z(x'). p d z = ϕ Z ( x ′ ) . This contradicts the minimality of d d d d = 1 d=1 d = 1 z z z ϕ Z \phi_Z ϕ Z ϕ Z \phi_Z ϕ Z
Finite and affine type A
To apply Theorem A(1)–(2), we need an upper bound on the dimension of Z ( R α ( k ) ) Z(R_{\alpha}(\mathbb{k})) Z ( R α ( k )) Theorem 4.5 below.
Let us recall the definition of non-degenerate cyclotomic Hecke algebras.
Definition 4.1 ([3 Def. 3.1] ). Let k k k q ∈ k × q \in k^{\times} q ∈ k × Q = ( Q 1 , … , Q r ) ∈ ( k × ) r \mathbf{Q}=(Q_1,\ldots,Q_r)\in(k^{\times})^r Q = ( Q 1 , … , Q r ) ∈ ( k × ) r H q , n Q = H q , n Q ( k ) \mathcal{H}^{\mathbf{Q}}_{q,n}=\mathcal{H}^{\mathbf{Q}}_{q,n}(k) H q , n Q = H q , n Q ( k ) G ( r , 1 , n ) G(r,1,n) G ( r , 1 , n ) q q q Q \mathbf{Q} Q k k k T 0 , T 1 , … , T n − 1 T_0,T_1,\ldots,T_{n-1} T 0 , T 1 , … , T n − 1
∏ m = 1 r ( T 0 − Q m ) = 0 , ( T a − q ) ( T a + 1 ) = 0 ( 1 ≤ a < n ) , \prod_{m=1}^{r}(T_0-Q_m)=0,\qquad(T_a-q)(T_a+1)=0\quad(1\leq a<n), m = 1 ∏ r ( T 0 − Q m ) = 0 , ( T a − q ) ( T a + 1 ) = 0 ( 1 ≤ a < n ) , T 0 T 1 T 0 T 1 = T 1 T 0 T 1 T 0 , T a T b = T b T a ( 0 ≤ a < b − 1 < n − 1 ) , T_0T_1T_0T_1=T_1T_0T_1T_0,\qquad T_aT_b=T_bT_a\quad(0\leq a<b-1<n-1), T 0 T 1 T 0 T 1 = T 1 T 0 T 1 T 0 , T a T b = T b T a ( 0 ≤ a < b − 1 < n − 1 ) , T a T a + 1 T a = T a + 1 T a T a + 1 ( 1 ≤ a < n − 1 ) . T_aT_{a+1}T_a=T_{a+1}T_aT_{a+1}\quad(1\leq a<n-1). T a T a + 1 T a = T a + 1 T a T a + 1 ( 1 ≤ a < n − 1 ) . The Jucys–Murphy elements of H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q L 1 = T 0 L_1=T_0 L 1 = T 0 L a + 1 = q − 1 T a L a T a L_{a+1}=q^{-1}T_aL_aT_a L a + 1 = q − 1 T a L a T a 1 ≤ a < n 1\leq a<n 1 ≤ a < n
Lemma 4.2 ([3 §3] ). The Jucys–Murphy elements L 1 , … , L n L_1,\ldots,L_n L 1 , … , L n H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q
We call the subalgebra Z ( H q , n Q ) J M ⊆ Z ( H q , n Q ) Z(\mathcal{H}^{\mathbf{Q}}_{q,n})^{\mathrm{JM}}\subseteq Z(\mathcal{H}^{\mathbf{Q}}_{q,n}) Z ( H q , n Q ) JM ⊆ Z ( H q , n Q ) L 1 , … , L n L_1,\ldots,L_n L 1 , … , L n H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q Z ( H q , n Q ) Z(\mathcal{H}^{\mathbf{Q}}_{q,n}) Z ( H q , n Q ) Z ( H q , n Q ) J M Z(\mathcal{H}^{\mathbf{Q}}_{q,n})^{\mathrm{JM}} Z ( H q , n Q ) JM k k k Z ( H q , n Q ) = Z ( H q , n Q ) J M Z(\mathcal{H}^{\mathbf{Q}}_{q,n})=Z(\mathcal{H}^{\mathbf{Q}}_{q,n})^{\mathrm{JM}} Z ( H q , n Q ) = Z ( H q , n Q ) JM H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q P ( T 0 ) = 0 P(T_0)=0 P ( T 0 ) = 0 P P P P ( 0 ) ≠ 0 P(0)\ne0 P ( 0 ) = 0
We first recall the extension of Brundan–Kleshchev’s isomorphism between cyclotomic KLR algebras and cyclotomic Hecke algebras to cyclotomic parameters lying in several q q q
Theorem 4.3 ([7 Main Theorem] , [Web20, Th. 1.6], [30 Th. 3.4] ). Let k k k q ∈ k × q\in k^{\times} q ∈ k × q ≠ 1 q\ne1 q = 1 Q ∈ ( k × ) r \mathbf{Q}\in(k^{\times})^r Q ∈ ( k × ) r H q , n Q ( k ) \mathcal{H}^{\mathbf{Q}}_{q,n}(k) H q , n Q ( k ) R n Λ ( k ) R_n^{\Lambda}(k) R n Λ ( k ) k k k Γ = ( I , E ) \Gamma=(I,E) Γ = ( I , E ) A ∞ A_{\infty} A ∞ A e − 1 ( 1 ) A_{e-1}^{(1)} A e − 1 ( 1 ) e ≥ 2 e\geq2 e ≥ 2
I = { q a Q m ∣ a ∈ Z , 1 ≤ m ≤ r } ⊂ k × , E = { ( x , q x ) ∣ x ∈ I } , (8) I=\{q^aQ_m\mid a\in\mathbb{Z},\ 1\leq m\leq r\}\subset k^{\times},\qquad E=\{(x,qx)\mid x\in I\}, \tag*{(8)} I = { q a Q m ∣ a ∈ Z , 1 ≤ m ≤ r } ⊂ k × , E = {( x , q x ) ∣ x ∈ I } , (8) and Λ = ∑ i ∈ I w i Λ i \Lambda=\sum_{i\in I}w_i\Lambda_i Λ = ∑ i ∈ I w i Λ i w i = # { m ∣ Q m = i } w_i=\#\{m\mid Q_m=i\} w i = # { m ∣ Q m = i }
If q q q e e e q q q I I I e e e Γ \Gamma Γ q q q A ∞ A_{\infty} A ∞ I I I I I I q q q [10 ] , and the KLR algebra makes the Morita equivalence of [10 Th. 1.2] with a sum of tensor products of Hecke algebras with q q q
We identify an r r r λ = ( λ ( 1 ) ∣ … ∣ λ ( r ) ) \boldsymbol{\lambda}=(\lambda^{(1)}|\ldots|\lambda^{(r)}) λ = ( λ ( 1 ) ∣ … ∣ λ ( r ) ) n n n a a a c c c m m m Q m q c − a ∈ I Q_mq^{c-a}\in I Q m q c − a ∈ I P n r \mathcal{P}_n^r P n r r r r n n n α = ∑ i v i α i ∈ Q + + \alpha=\sum_i v_i\alpha_i\in Q_+^+ α = ∑ i v i α i ∈ Q + + P α r \mathcal{P}_{\alpha}^r P α r λ ∈ P n r \boldsymbol{\lambda}\in\mathcal{P}_n^r λ ∈ P n r v i v_i v i i i i i ∈ I i\in I i ∈ I P n r = ⋃ α ∈ Q + + P α r \mathcal{P}_n^r=\bigcup_{\alpha\in Q_+^+}\mathcal{P}_{\alpha}^r P n r = ⋃ α ∈ Q + + P α r
Lemma 4.4. For any field k k k Γ \Gamma Γ Theorem 4.3 , we have dim H ∗ ( M Λ , α ; k ) = ∣ P α r ∣ \dim H^*(\mathcal{M}_{\Lambda,\alpha};k)=|\mathcal{P}_{\alpha}^r| dim H ∗ ( M Λ , α ; k ) = ∣ P α r ∣
Proof. Since the quiver variety for a union of quivers is the product of those for its components, we may assume Γ \Gamma Γ T = ( C × ) 2 × T w T=(\mathbb{C}^{\times})^2\times T_w T = ( C × ) 2 × T w M Λ , α \mathcal{M}_{\Lambda,\alpha} M Λ , α C 2 \mathbb{C}^2 C 2 P α r \mathcal{P}_{\alpha}^r P α r [12 Prop. 3.8] . Note that Fujii–Minabe’s approach only applies precisely as written if I = μ e I=\mu_e I = μ e e e e Q m q c − a Q_mq^{c-a} Q m q c − a q ∈ μ e q\in\mu_e q ∈ μ e Γ \Gamma Γ e e e e e e Γ \Gamma Γ A ∞ A_\infty A ∞ e e e e e e
The Białynicki-Birula decomposition for an appropriately chosen cocharacter of T T T M ( v , w ) \mathcal{M}(\mathbf{v},\mathbf{w}) M ( v , w ) [27 Lem. 7.1.5] . Thus H ∗ ( M Λ , α ; Z ) H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z}) H ∗ ( M Λ , α ; Z ) ∣ P α ∣ |\mathscr{P}_{\alpha}| ∣ P α ∣ dim H ∗ ( M Λ , α ; k ) = ∣ P α ∣ \dim H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{k})=|\mathscr{P}_{\alpha}| dim H ∗ ( M Λ , α ; k ) = ∣ P α ∣ k \mathbb{k} k
We use the following dimension theorem.
Theorem 4.5 ([15 Th. 1.4] , [6 Th. 1] ). Let k \mathbb{k} k q ≠ 1 q\ne1 q = 1 dim Z ( H q , n Q ( k ) ) = ∣ P n r ∣ \dim Z(\mathcal{H}^{\mathbf{Q}}_{q,n}(\mathbb{k}))=|\mathscr{P}^{r}_{n}| dim Z ( H q , n Q ( k )) = ∣ P n r ∣ r r r
Proof of Theorem A, part (3). We may extend the field, since the relevant dimensions are unchanged by field extension. If p = 0 p=0 p = 0 p ∤ e p\nmid e p ∤ e e e e q ∈ k q\in\mathbb{k} q ∈ k Q \mathbf{Q} Q q q q Γ \Gamma Γ Λ \Lambda Λ R n Λ ( k ) R^{\Lambda}_{n}(\mathbb{k}) R n Λ ( k ) H q , n Q ( k ) \mathcal{H}^{\mathbf{Q}}_{q,n}(\mathbb{k}) H q , n Q ( k ) p = e p=e p = e [7 ] gives the same identification with a degenerate cyclotomic Hecke algebra.
By Theorem A(1), the direct sum
ϕ k : ⨁ α ∈ Q n + H ∗ ( M Λ , α ; k ) ⟶ ⨁ α ∈ Q n + Z ( R α Λ ( k ) ) = Z ( R n Λ ( k ) ) \phi_{\mathbb{k}}:\bigoplus_{\alpha\in Q^{+}_{n}}H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{k})\longrightarrow\bigoplus_{\alpha\in Q^{+}_{n}}Z(R^{\Lambda}_{\alpha}(\mathbb{k}))=Z(R^{\Lambda}_{n}(\mathbb{k})) ϕ k : α ∈ Q n + ⨁ H ∗ ( M Λ , α ; k ) ⟶ α ∈ Q n + ⨁ Z ( R α Λ ( k )) = Z ( R n Λ ( k )) is injective. By Lemma 4.4, its source has dimension ∣ P n r ∣ |\mathscr{P}^{r}_{n}| ∣ P n r ∣ H ∗ ( M Λ , α ; k ) → Z ( R α Λ ( k ) ) H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{k})\to Z(R^{\Lambda}_{\alpha}(\mathbb{k})) H ∗ ( M Λ , α ; k ) → Z ( R α Λ ( k ))
Applying Proposition 3.12 to the characteristic-zero case of Theorem A(3) gives the following integral isomorphism, with no restriction on the prime divisors of e e e
Corollary 4.6. Suppose that every connected component of Γ \Gamma Γ e e e e ≥ 2 e\ge2 e ≥ 2 Λ \Lambda Λ α ∈ Q + \alpha\in Q^{+} α ∈ Q +
ϕ Z : H ∗ ( M Λ , α ; Z ) ⟶ Z ( R α Λ ( Z ) ) \phi_{\mathbb{Z}}:H^*(\mathcal{M}_{\Lambda,\alpha};\mathbb{Z})\longrightarrow Z(R^{\Lambda}_{\alpha}(\mathbb{Z})) ϕ Z : H ∗ ( M Λ , α ; Z ) ⟶ Z ( R α Λ ( Z )) is an isomorphism of rings.
Proof. By Theorem A(3), applied over Q \mathbb{Q} Q ϕ Q \phi_{\mathbb{Q}} ϕ Q
Applying Theorem A(1) on the other side of the Brundan–Kleshchev isomorphism of Theorem 4.3, we obtain the proof of Theorem B.
Proof of Theorem B. Consider the subset I ⊂ k × I\subset\mathbb{k}^{\times} I ⊂ k × (8) and the graph Γ \Gamma Γ α = ∑ i v i α i ∈ Q n + \alpha=\sum_i v_i\alpha_i\in Q^{+}_{n} α = ∑ i v i α i ∈ Q n + H q , α Q \mathcal{H}^{\mathbf{Q}}_{q,\alpha} H q , α Q H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q L 1 , … , L n L_1,\ldots,L_n L 1 , … , L n i ∈ I i\in I i ∈ I v i v_i v i H q , α Q \mathcal{H}^{\mathbf{Q}}_{q,\alpha} H q , α Q R α Λ R^{\Lambda}_{\alpha} R α Λ H q , n Q = ⨁ α H q , α Q \mathcal{H}^{\mathbf{Q}}_{q,n}=\bigoplus_{\alpha}\mathcal{H}^{\mathbf{Q}}_{q,\alpha} H q , n Q = ⨁ α H q , α Q n i ; j ∈ Z ( H q , α Q ) n_{i;j}\in Z(\mathcal{H}^{\mathbf{Q}}_{q,\alpha}) n i ; j ∈ Z ( H q , α Q ) j j j L a L_a L a i i i n i ; j n_{i;j} n i ; j i j e i ; j i^je_{i;j} i j e i ; j b ( u ) = 1 + u b(u)=1+u b ( u ) = 1 + u L a e ( i ) L_a e(i) L a e ( i ) i a ( 1 + y a ) e ( i ) i_a(1+y_a)e(i) i a ( 1 + y a ) e ( i )
We now write each n i ; j n_{i;j} n i ; j L a L_a L a C ⊂ I C\subset I C ⊂ I L 1 , … , L n L_1,\ldots,L_n L 1 , … , L n H q , n Q \mathcal{H}^{\mathbf{Q}}_{q,n} H q , n Q N N N ( L a − i d ) N e ( i ) = 0 (L_a-i_d)^N e(i)=0 ( L a − i d ) N e ( i ) = 0 a a a i i i i ∈ C i\in C i ∈ C ℓ i ( T ) = ∏ c ∈ C ∖ { i } T − c i − c \ell_i(T)=\prod_{c\in C\setminus\{i\}}\frac{T-c}{i-c} ℓ i ( T ) = ∏ c ∈ C ∖ { i } i − c T − c
p i ( T ) = 1 − ( 1 − ℓ i ( T ) N ) N . p_i(T)=1-\left(1-\ell_i(T)^N\right)^N. p i ( T ) = 1 − ( 1 − ℓ i ( T ) N ) N . Then
p i ( T ) ≡ δ i c ( m o d ( T − c ) N ) ( c ∈ C ) . p_i(T)\equiv\delta_{ic}\pmod{(T-c)^N}\qquad(c\in C). p i ( T ) ≡ δ i c ( mod ( T − c ) N ) ( c ∈ C ) . Indeed, we have 1 − ℓ i ( T ) N ≡ 1 ( m o d ( T − c ) N ) 1-\ell_i(T)^N\equiv1\pmod{(T-c)^N} 1 − ℓ i ( T ) N ≡ 1 ( mod ( T − c ) N ) c ≠ i c\ne i c = i T − i T-i T − i 1 − ℓ i N 1-\ell_i^N 1 − ℓ i N ( T − i ) N (T-i)^N ( T − i ) N ( 1 − ℓ i N ) N (1-\ell_i^N)^N ( 1 − ℓ i N ) N D i ( T ) = ( T − i ) p i ( T ) D_i(T)=(T-i)p_i(T) D i ( T ) = ( T − i ) p i ( T )
p i ( L a ) e ( i ) = { e ( i ) , i a = i , 0 , i a ≠ i , D i ( L a ) e ( i ) = { ( L a − i ) e ( i ) , i a = i , 0 , i a ≠ i . (9) p_i(L_a)e(\mathbf{i})=
\begin{cases}
e(\mathbf{i}), & i_a=i,\\
0, & i_a\ne i,
\end{cases}
\qquad
D_i(L_a)e(\mathbf{i})=
\begin{cases}
(L_a-i)e(\mathbf{i}), & i_a=i,\\
0, & i_a\ne i.
\end{cases} \tag*{(9)} p i ( L a ) e ( i ) = { e ( i ) , 0 , i a = i , i a = i , D i ( L a ) e ( i ) = { ( L a − i ) e ( i ) , 0 , i a = i , i a = i . (9) This means that p i 1 ( L 1 ) ⋯ p i n ( L n ) p_{i_1}(L_1)\cdots p_{i_n}(L_n) p i 1 ( L 1 ) ⋯ p i n ( L n ) e ( i ) e(\mathbf{i}) e ( i ) i = ( i 1 , … , i n ) ∈ C n \mathbf{i}=(i_1,\ldots,i_n)\in C^n i = ( i 1 , … , i n ) ∈ C n e ( i ) = 0 e(\mathbf{i})=0 e ( i ) = 0 i ∉ C n \mathbf{i}\notin C^n i ∈ / C n ∑ i ∈ I α ∩ C n p i 1 ( L 1 ) ⋯ p i n ( L n ) = 1 α \sum_{\mathbf{i}\in I^\alpha\cap C^n}p_{i_1}(L_1)\cdots p_{i_n}(L_n)=1_\alpha ∑ i ∈ I α ∩ C n p i 1 ( L 1 ) ⋯ p i n ( L n ) = 1 α H q , α Q \mathcal{H}_{q,\alpha}^{\mathbf{Q}} H q , α Q 1 α ∈ Z ( H q , α Q ) J M 1_\alpha\in Z(\mathcal{H}_{q,\alpha}^{\mathbf{Q}})^{\mathrm{JM}} 1 α ∈ Z ( H q , α Q ) JM Z ( H q , n Q ) J M = ⨁ α 1 α Z ( H q , n Q ) J M Z(\mathcal{H}_{q,n}^{\mathbf{Q}})^{\mathrm{JM}}=\bigoplus_\alpha1_\alpha Z(\mathcal{H}_{q,n}^{\mathbf{Q}})^{\mathrm{JM}} Z ( H q , n Q ) JM = ⨁ α 1 α Z ( H q , n Q ) JM
Under the above identification, for each i ∈ I α \mathbf{i}\in I^\alpha i ∈ I α (9) gives
e j ( D i ( L 1 ) , … , D i ( L n ) ) e ( i ) = ∑ 1 ≤ a 1 < ⋯ < a j ≤ n i a 1 = ⋯ = i a j = i ( L a 1 − i ) ⋯ ( L a j − i ) e ( i ) = i j e j ( y a ∣ i a = i ) e ( i ) = n i ; j e ( i ) . (10) e_j(D_i(L_1),\ldots,D_i(L_n))e(\mathbf{i})
=
\sum_{\substack{1\leq a_1<\cdots<a_j\leq n\\i_{a_1}=\cdots=i_{a_j}=i}}
(L_{a_1}-i)\cdots(L_{a_j}-i)e(\mathbf{i})
= i^j e_j(y_a\mid i_a=i)e(\mathbf{i})
= n_{i;j}e(\mathbf{i}). \tag*{(10)} e j ( D i ( L 1 ) , … , D i ( L n )) e ( i ) = 1 ≤ a 1 < ⋯ < a j ≤ n i a 1 = ⋯ = i a j = i ∑ ( L a 1 − i ) ⋯ ( L a j − i ) e ( i ) = i j e j ( y a ∣ i a = i ) e ( i ) = n i ; j e ( i ) . (10) The second equality uses ( L a − i ) e ( i ) = i y a e ( i ) (L_a-i)e(\mathbf{i})=i y_a e(\mathbf{i}) ( L a − i ) e ( i ) = i y a e ( i ) i a = i i_a=i i a = i n i ; j n_{i;j} n i ; j e ( i ) = 0 e(\mathbf{i})=0 e ( i ) = 0
Since ∑ i ∈ I α e ( i ) = 1 \sum_{\mathbf{i}\in I^\alpha}e(\mathbf{i})=1 ∑ i ∈ I α e ( i ) = 1 H q , α Q \mathcal{H}_{q,\alpha}^{\mathbf{Q}} H q , α Q (10) gives
e j ( D i ( L 1 ) , … , D i ( L n ) ) = ∑ i ∈ I α e j ( D i ( L 1 ) , … , D i ( L n ) ) e ( i ) = i j ∑ i ∈ I α e j ( y a ∣ i a = i ) e ( i ) = i j e i ; j = n i ; j . (11) e_j(D_i(L_1),\ldots,D_i(L_n))
=
\sum_{\mathbf{i}\in I^\alpha}e_j(D_i(L_1),\ldots,D_i(L_n))e(\mathbf{i})
= i^j\sum_{\mathbf{i}\in I^\alpha}e_j(y_a\mid i_a=i)e(\mathbf{i})
= i^j e_{i;j}=n_{i;j}. \tag*{(11)} e j ( D i ( L 1 ) , … , D i ( L n )) = i ∈ I α ∑ e j ( D i ( L 1 ) , … , D i ( L n )) e ( i ) = i j i ∈ I α ∑ e j ( y a ∣ i a = i ) e ( i ) = i j e i ; j = n i ; j . (11) So n i ; j = e j ( D i ( L 1 ) , … , D i ( L n ) ) 1 α n_{i;j}=e_j(D_i(L_1),\ldots,D_i(L_n))1_\alpha n i ; j = e j ( D i ( L 1 ) , … , D i ( L n )) 1 α Z ( H q , α Q ) J M Z(\mathcal{H}_{q,\alpha}^{\mathbf{Q}})^{\mathrm{JM}} Z ( H q , α Q ) JM
Conversely, on a fixed summand H q , α Q \mathcal{H}_{q,\alpha}^{\mathbf{Q}} H q , α Q L a L_a L a i i i n i ; j n_{i;j} n i ; j n i ; j n_{i;j} n i ; j Z ( H q , α Q ) J M Z(\mathcal{H}_{q,\alpha}^{\mathbf{Q}})^{\mathrm{JM}} Z ( H q , α Q ) JM
S → κ g Z ( R α ( k ) ) → κ H Z ( H q , α Q ) ϕ κ ↑ ↑ ∼ H ∗ ( M ( v , w ) ; k ) → Z ( R α ( k ) ) → Z ( H q , α Q ) \begin{CD}
S @>{\kappa_g}>> Z(R_\alpha(\mathbb{k})) @>{\kappa_H}>> Z(\mathcal{H}_{q,\alpha}^{\mathbf{Q}})\\
@. @A{\phi_\kappa}AA @AA{\sim}A\\
H^*(\mathcal{M}(\mathbf{v},\mathbf{w});\mathbb{k}) @>>> Z(R_\alpha(\mathbb{k})) @>>> Z(\mathcal{H}_{q,\alpha}^{\mathbf{Q}})
\end{CD} S H ∗ ( M ( v , w ) ; k ) κ g Z ( R α ( k )) ϕ κ ⏐ ↑ Z ( R α ( k )) κ H Z ( H q , α Q ) ⏐ ↑ ∼ Z ( H q , α Q ) where, by definition, the map κ H \kappa_H κ H c i ; j c_{i;j} c i ; j i − j n i ; j i^{-j}n_{i;j} i − j n i ; j κ g \kappa_g κ g κ H \kappa_H κ H Z ( H q , α Q ) J M Z(\mathcal{H}_{q,\alpha}^{\mathbf{Q}})^{\mathrm{JM}} Z ( H q , α Q ) JM ∣ P α ∣ |\mathscr{P}_\alpha| ∣ P α ∣ α \alpha α α ∈ Q n + \alpha\in Q_n^+ α ∈ Q n +
dim Z ( H q , n Q ) > ∑ α ∈ Q n + ∣ P α ∣ = ∣ P n ∣ , \dim Z(\mathcal{H}_{q,n}^{\mathbf{Q}})>\sum_{\alpha\in Q_n^+}|\mathscr{P}_\alpha|=|\mathscr{P}_n|, dim Z ( H q , n Q ) > α ∈ Q n + ∑ ∣ P α ∣ = ∣ P n ∣ , contradicting Theorem 4.5. Thus κ H \kappa_H κ H α \alpha α Z ( H q , n Q ) = Z ( H q , n Q ) J M Z(\mathcal{H}_{q,n}^{\mathbf{Q}})=Z(\mathcal{H}_{q,n}^{\mathbf{Q}})^{\mathrm{JM}} Z ( H q , n Q ) = Z ( H q , n Q ) JM
Careful investigation of the proof of Theorem B reveals that the proof does not use the classification of the blocks of H q , n Q \mathcal{H}_{q,n}^{\mathbf{Q}} H q , n Q [22 ] . In several places the proof uses Lemma 4.2, but this follows easily from the relations. Following [9 Def. 3.10, Th. 3.26] , H q , n Q \mathcal{H}_{q,n}^{\mathbf{Q}} H q , n Q S λ S^\lambda S λ λ ∈ P n \lambda\in\mathscr{P}_n λ ∈ P n
Corollary 4.7 (Lyle–Mathas [22 Th. A] ). Let k k k q ≠ 1 q\ne1 q = 1 Q ∈ ( k × ) r \mathbf{Q}\in(k^\times)^r Q ∈ ( k × ) r α ∈ Q n + \alpha\in Q_n^+ α ∈ Q n + P α ≠ ∅ \mathscr{P}_\alpha\ne\varnothing P α = ∅ H q , α Q \mathcal{H}_{q,\alpha}^{\mathbf{Q}} H q , α Q H q , n Q \mathcal{H}_{q,n}^{\mathbf{Q}} H q , n Q S λ S^\lambda S λ H q , α Q \mathcal{H}_{q,\alpha}^{\mathbf{Q}} H q , α Q λ ∈ P α \lambda\in\mathscr{P}_\alpha λ ∈ P α S λ S^\lambda S λ S μ S^\mu S μ λ \lambda λ μ \mu μ
Proof. We have now proven that Z ( H q , α Q ) ≅ Z ( R α ( k ) ) J M Z(\mathcal{H}_{q,\alpha}^{\mathbf{Q}})\cong Z(R_\alpha(\mathbb{k}))^{\mathrm{JM}} Z ( H q , α Q ) ≅ Z ( R α ( k ) ) JM H q , α Q \mathcal{H}_{q,\alpha}^{\mathbf{Q}} H q , α Q L a L_a L a [17 Prop. 3.7] . □
Remark 4.8. Suppose p ∣ e p \mid e p ∣ e p ≠ e p \ne e p = e p p p A e − 1 ( 1 ) A_{e-1}^{(1)} A e − 1 ( 1 ) ξ ∈ F ‾ p \xi\in\overline{\mathbb{F}}_p ξ ∈ F p p p p x ↦ x + 1 x \mapsto x+1 x ↦ x + 1 p p p Theorem 4.5 is not available. By Theorem A, ϕ k \phi_k ϕ k dim Z ( R α Λ ( k ) ) ≥ ∣ P α Λ ∣ \dim Z(R_{\alpha}^{\Lambda}(k)) \geq|\mathcal{P}_{\alpha}^{\Lambda}| dim Z ( R α Λ ( k )) ≥ ∣ P α Λ ∣ R α Λ ( k ) R_{\alpha}^{\Lambda}(k) R α Λ ( k ) Lemma 3.7 ), dim Z ( R α Λ ( k ) ) = dim Tr ( R α Λ ( k ) ) = dim Tr ( R α Λ ) ⊗ k \dim Z(R_{\alpha}^{\Lambda}(k)) = \dim\operatorname{Tr}(R_{\alpha}^{\Lambda}(k)) = \dim\operatorname{Tr}(R_{\alpha}^{\Lambda}) \otimes k dim Z ( R α Λ ( k )) = dim Tr ( R α Λ ( k )) = dim Tr ( R α Λ ) ⊗ k ∣ P α Λ ∣ |\mathcal{P}_{\alpha}^{\Lambda}| ∣ P α Λ ∣ p p p Tr ( R α Λ ) \operatorname{Tr}(R_{\alpha}^{\Lambda}) Tr ( R α Λ ) ϕ k \phi_k ϕ k Tr ( R α Λ ) \operatorname{Tr}(R_{\alpha}^{\Lambda}) Tr ( R α Λ ) p p p
The center and K K K
This final section proves Theorem C , which connects the K K K [21 , 29 ] .
We begin with an equivariant form of Kirwan surjectivity in K K K M = M ( v , w ) \mathcal{M} = \mathcal{M}(\mathbf{v},\mathbf{w}) M = M ( v , w ) § 2.1 , but allowing edge loops. The quiver variety M \mathcal{M} M G w = ∏ i G L ( W i ) G_{\mathbf{w}} = \prod_{i} GL(W_i) G w = ∏ i G L ( W i ) ( G m ) E (\mathbb{G}_m)^E ( G m ) E h h h h ˉ \bar h h ˉ S = G m S = \mathbb{G}_m S = G m
⨁ h ∈ E ∪ E ‾ Hom ( V out ( h ) , V in ( h ) ) ⊕ ⨁ i ∈ I ( Hom ( V i , W i ) ⊕ Hom ( W i , V i ) ) . \bigoplus_{h \in E \cup\overline{E}} \operatorname{Hom}(V_{\operatorname{out}(h)},V_{\operatorname{in}(h)}) \oplus\bigoplus_{i \in I}\left(\operatorname{Hom}(V_i,W_i) \oplus\operatorname{Hom}(W_i,V_i)\right). h ∈ E ∪ E ⨁ Hom ( V out ( h ) , V in ( h ) ) ⊕ i ∈ I ⨁ ( Hom ( V i , W i ) ⊕ Hom ( W i , V i ) ) . We’ll be interested in a torus T T T M \mathcal{M} M T → G w × ( G m ) E × S T \to G_{\mathbf{w}} \times(\mathbb{G}_m)^E \times S T → G w × ( G m ) E × S V i \mathcal{V}_i V i M \mathcal{M} M T T T V i V_i V i T T T G v G_{\mathbf{v}} G v standard action on this vector bundle.
For a torus T T T X X X K T ( X ) K_T(X) K T ( X ) T T T X X X
Lemma 5.1. For T T T R ( G v × T ) → K T ( M ) R(G_{\mathbf{v}} \times T) \to K_T(\mathcal{M}) R ( G v × T ) → K T ( M ) T T T K ( M ) K(\mathcal{M}) K ( M ) (12) is generated as a ring by the classes [ Λ k V i ] [\Lambda^k\mathcal{V}_i] [ Λ k V i ] [ Λ v i V i ] − 1 [\Lambda^{v_i}\mathcal{V}_i]^{-1} [ Λ v i V i ] − 1 i ∈ I i \in I i ∈ I 1 ≤ k ≤ v i 1 \leq k \leq v_i 1 ≤ k ≤ v i
Proof. Without the torus, the first statement follows from [24 Th. 1.6(2)] : D b Coh ( M ) D^b\operatorname{Coh}(\mathcal{M}) D b Coh ( M ) T T T [24 p. 163] , and we use their notation.
The image of T T T G w G_{\mathbf{w}} G w W i W_i W i i i i j j j Q Q Q [24 §3.2] , with dimension vector α \alpha α v \mathbf{v} v I I I 1 1 1 ∞ \infty ∞ G ( α ) ≅ G v G(\alpha) \cong G_{\mathbf{v}} G ( α ) ≅ G v η : T → S \eta:T \to S η : T → S T → G w × ( G m ) E × S T \to G_{\mathbf{w}} \times(\mathbb{G}_m)^E \times S T → G w × ( G m ) E × S T T T h h h χ h \chi_h χ h χ h χ h ˉ = η 2 \chi_h\chi_{\bar h}=\eta^2 χ h χ h ˉ = η 2 G w G_{\mathbf{w}} G w ( G m ) E (\mathbb{G}_m)^E ( G m ) E h h h h ˉ \bar h h ˉ S S S η \eta η
First, we note that the compactification M ‾ \overline{\mathcal{M}} M T T T t ∈ T t \in T t ∈ T Rep ( Q gtr , α gtr ) \operatorname{Rep}(Q^{\operatorname{gtr}},\alpha^{\operatorname{gtr}}) Rep ( Q gtr , α gtr ) a h , n ↦ χ h ( t ) a h , n a_{h,n} \mapsto\chi_h(t)a_{h,n} a h , n ↦ χ h ( t ) a h , n e i , n ↦ e i , n e_{i,n} \mapsto e_{i,n} e i , n ↦ e i , n [24 Def. 4.3] : it multiplies the preprojective relations by η ( t ) 2 \eta(t)^2 η ( t ) 2 e a = a e ea = ae e a = a e a h , n a_{h,n} a h , n G gtr G^{\operatorname{gtr}} G gtr θ gtr \theta^{\operatorname{gtr}} θ gtr T T T M ‾ \overline{\mathcal{M}} M M \mathcal{M} M ι = Ind : M → M ‾ \iota=\operatorname{Ind}:\mathcal{M}\to\overline{\mathcal{M}} ι = Ind : M → M T T T
Second, consider the Koszul resolution of the graph Δ M ⊂ M × M ‾ \Delta_{\mathcal{M}}\subset\mathcal{M}\times\overline{\mathcal{M}} Δ M ⊂ M × M ι \iota ι T T T L ( V ∞ , 0 , W ∞ , 0 ) L(V_{\infty,0},W_{\infty,0}) L ( V ∞ , 0 , W ∞ , 0 ) χ h \chi_h χ h E ( V ∞ , 0 , W ∞ , 1 ) E(V_{\infty,0},W_{\infty,1}) E ( V ∞ , 0 , W ∞ , 1 ) h h h η 2 \eta^2 η 2 L ( V ∞ , 0 , W ∞ , 2 ) L(V_{\infty,0},W_{\infty,2}) L ( V ∞ , 0 , W ∞ , 2 ) ∂ 0 , ∂ 1 \partial_0,\partial_1 ∂ 0 , ∂ 1 [24 (5.3)] are T T T h h h ∂ 0 ( ϕ ) \partial_0(\phi) ∂ 0 ( ϕ ) χ h \chi_h χ h ∂ 1 ( ψ ) \partial_1(\psi) ∂ 1 ( ψ ) ψ h \psi_h ψ h χ h ˉ \chi_{\bar h} χ h ˉ η 2 \eta^2 η 2 s s s 0 0 0 1 ∈ Hom ( V ∞ , 0 , W ∞ , 0 ) 1 \in\operatorname{Hom}(V_{\infty,0},W_{\infty,0}) 1 ∈ Hom ( V ∞ , 0 , W ∞ , 0 ) T T T [24 (6.2)] of O Δ M \mathcal{O}_{\Delta_\mathcal{M}} O Δ M T T T [24 Th. 1.6] , this exhibits O Δ M \mathcal{O}_{\Delta_\mathcal{M}} O Δ M D b Coh T ( M × M ‾ ) D^b\operatorname{Coh}^T(\mathcal{M}\times\overline{\mathcal{M}}) D b Coh T ( M × M ) A ⊠ B A\boxtimes B A ⊠ B A A A Λ ( E j ) \Lambda(E_j) Λ ( E j ) B B B S μ ( F j ) S_\mu(F_j) S μ ( F j ) E j E_j E j M \mathcal{M} M F j F_j F j M ‾ \overline{\mathcal{M}} M ( A ⊗ χ ) ⊠ B = A ⊠ ( B ⊗ χ ) (A\otimes\chi)\boxtimes B=A\boxtimes(B\otimes\chi) ( A ⊗ χ ) ⊠ B = A ⊠ ( B ⊗ χ ) χ \chi χ T T T T T T A A A B B B
Now let F F F T T T M ‾ \overline{\mathcal{M}} M p 1 , p 2 p_1,p_2 p 1 , p 2 M × M ‾ \mathcal{M}\times\overline{\mathcal{M}} M × M ι \iota ι ι ∗ F ≅ R p 1 ∗ ( O Δ M ⊗ L p 2 ∗ F ) \iota^*F \cong Rp_{1*}(\mathcal{O}_{\Delta_\mathcal{M}}\otimes^L p_2^*F) ι ∗ F ≅ R p 1 ∗ ( O Δ M ⊗ L p 2 ∗ F ) p 1 p_1 p 1 M ‾ \overline{\mathcal{M}} M ι ∗ F \iota^*F ι ∗ F D b Coh T ( M ) D^b\operatorname{Coh}^T(\mathcal{M}) D b Coh T ( M ) A ⊗ U A\otimes U A ⊗ U U U U T T T H k ( M ‾ , F ⊗ B ) H^k(\overline{\mathcal{M}},\mathcal{F}\otimes B) H k ( M , F ⊗ B ) [ ι ∗ F ] [\iota^*\mathcal{F}] [ ι ∗ F ] [ A ] [A] [ A ] T T T K T ( p t ) = R ( T ) K_T(\mathrm{pt})=R(T) K T ( pt ) = R ( T ) ι ∗ : K T ( M ‾ ) → K T ( M ) \iota^*:K_T(\overline{\mathcal{M}})\to K_T(\mathcal{M}) ι ∗ : K T ( M ) → K T ( M ) ι ∗ \iota^* ι ∗ [8 §5.2.14] , the Kirwan map is surjective.
The last statement holds because R ( G v ) R(G_{\mathbf{v}}) R ( G v ) V i V_i V i
Following [21 Section 2.1 and Definition 3.1] , but working over Z \mathbb{Z} Z
A = Z [ q ± 1 , Q 1 ± 1 , … , Q r ± 1 ] , and A ∘ = A [ ( q − 1 ) − 1 ] , \mathcal{A}=\mathbb{Z}[q^{\pm1},Q_1^{\pm1},\ldots,Q_r^{\pm1}],
\qquad\text{and}\qquad
\mathcal{A}^{\circ}=\mathcal{A}[(q-1)^{-1}], A = Z [ q ± 1 , Q 1 ± 1 , … , Q r ± 1 ] , and A ∘ = A [( q − 1 ) − 1 ] , and consider the Hecke algebra H q , n Q = H q , n Q ( A ∘ ) \mathcal{H}^{\mathsf{Q}}_{q,n}=\mathcal{H}^{\mathsf{Q}}_{q,n}(\mathcal{A}^{\circ}) H q , n Q = H q , n Q ( A ∘ ) A ∘ \mathcal{A}^{\circ} A ∘ q q q Q Q Q A C = C ⊗ Z A \mathcal{A}_{\mathbb{C}}=\mathbb{C}\otimes_{\mathbb{Z}}\mathcal{A} A C = C ⊗ Z A A C ∘ = C ⊗ Z A ∘ \mathcal{A}^{\circ}_{\mathbb{C}}=\mathbb{C}\otimes_{\mathbb{Z}}\mathcal{A}^{\circ} A C ∘ = C ⊗ Z A ∘ [21 ] .
Let G ( n , r ) \mathcal{G}(n,r) G ( n , r ) n n n r r r [28 ] , G ( n , r ) \mathcal{G}(n,r) G ( n , r ) P 2 ( C ) \mathbb{P}^2(\mathbb{C}) P 2 ( C ) r r r n n n L ( n , r ) \mathcal{L}(n,r) L ( n , r ) 0 0 0 G ( n , r ) → G 0 ( n , r ) \mathcal{G}(n,r)\to\mathcal{G}_0(n,r) G ( n , r ) → G 0 ( n , r ) T = T × T r \mathcal{T}=\mathbb{T}\times\mathbb{T}_r T = T × T r T \mathbb{T} T S L 2 SL_2 S L 2 T r \mathbb{T}_r T r G L r GL_r G L r K T ( X ) K_{\mathcal{T}}(X) K T ( X ) T \mathcal{T} T X X X K T ( p t ) = A K_{\mathcal{T}}(\mathrm{pt})=\mathcal{A} K T ( pt ) = A K T ( X ; C ) = C ⊗ Z K T ( X ) K_{\mathcal{T}}(X;\mathbb{C})=\mathbb{C}\otimes_{\mathbb{Z}}K_{\mathcal{T}}(X) K T ( X ; C ) = C ⊗ Z K T ( X )
Krylov, Paegelow and Shlykov [21 Theorem 3.16] proved that the equivariant K K K H q , n Q ( A C ) \mathcal{H}^{\mathsf{Q}}_{q,n}(\mathcal{A}_{\mathbb{C}}) H q , n Q ( A C ) K K K K T ( G ( n , r ) ; C ) ≅ Z ( H q , n Q ( A C ) ) J M K_{\mathcal{T}}(\mathcal{G}(n,r);\mathbb{C})\cong Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(\mathcal{A}_{\mathbb{C}}))^{\mathrm{JM}} K T ( G ( n , r ) ; C ) ≅ Z ( H q , n Q ( A C ) ) JM i i i e i ( L 1 , … , L n ) e_i(L_1,\ldots,L_n) e i ( L 1 , … , L n )
There exists a perfect pairing ⟨ − , − ⟩ \langle-,-\rangle ⟨ − , − ⟩ K T ( G ( n , r ) ; C ) K_{\mathcal{T}}(\mathcal{G}(n,r);\mathbb{C}) K T ( G ( n , r ) ; C ) K T ( L ( n , r ) ; C ) K_{\mathcal{T}}(\mathcal{L}(n,r);\mathbb{C}) K T ( L ( n , r ) ; C ) [21 (4.1)] , and by [21 Lemma 4.5] there is a unique isomorphism
ψ : Frac ( A C ) ⊗ A K T ( L ( n , r ) ) ⟶ Frac ( A C ) ⊗ A Tr ( H q , n Q ( A ) ) \psi:\operatorname{Frac}(\mathcal{A}_{\mathbb{C}})\otimes_{\mathcal{A}}K_{\mathcal{T}}(\mathcal{L}(n,r))\longrightarrow\operatorname{Frac}(\mathcal{A}_{\mathbb{C}})\otimes_{\mathcal{A}}\operatorname{Tr}(\mathcal{H}^{\mathsf{Q}}_{q,n}(\mathcal{A})) ψ : Frac ( A C ) ⊗ A K T ( L ( n , r )) ⟶ Frac ( A C ) ⊗ A Tr ( H q , n Q ( A )) of Z ( H q , n Q ( A C ) ) Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(\mathcal{A}_{\mathbb{C}})) Z ( H q , n Q ( A C )) ( z , [ h ] ) ↦ τ ( z h ) (z,[h])\mapsto\tau(zh) ( z , [ h ]) ↦ τ ( z h ) τ \tau τ H q , n Q \mathcal{H}^{\mathsf{Q}}_{q,n} H q , n Q [15 Lemma 2.6] . Krylov, Paegelow and Shlykov formulated the following conjecture, which asserts that ψ \psi ψ [21 Section 4] , q ≠ 1 q\ne1 q = 1 q = 1 q=1 q = 1
Conjecture 5.2 (Compare [21 Conjecture 4.6] ). There is a natural Z ( H q , n ∘ ) Z(\mathcal{H}^{\circ}_{q,n}) Z ( H q , n ∘ ) A ∘ ⊗ A K T ( L ( n , r ) ) ≅ Tr ( H q , n ∘ ) \mathcal{A}^{\circ}\otimes_{\mathcal{A}}K_{\mathcal{T}}(\mathcal{L}(n,r))\cong\operatorname{Tr}(\mathcal{H}^{\circ}_{q,n}) A ∘ ⊗ A K T ( L ( n , r )) ≅ Tr ( H q , n ∘ )
The Gieseker variety is the quiver variety of the Jordan quiver with v = n \mathbf{v}=n v = n w = r \mathbf{w}=r w = r G v = G L n G_{\mathbf{v}}=GL_n G v = G L n G w = G L r G_{\mathbf{w}}=GL_r G w = G L r E E E ( G m ) E (G_m)^E ( G m ) E T \mathbb{T} T S L 2 SL_2 S L 2 B 1 B_1 B 1 B 2 B_2 B 2 T = T × T r \mathcal{T}=\mathbb{T}\times\mathbb{T}_r T = T × T r ( G m ) E × G w (G_m)^E\times G_{\mathbf{w}} ( G m ) E × G w S S S Lemma 5.1 shows that the Kirwan map R ( G L n × T ) → K T ( G ( n , r ) ) R(GL_n\times\mathbb{T})\to K_{\mathcal{T}}(\mathcal{G}(n,r)) R ( G L n × T ) → K T ( G ( n , r )) C \mathbb{C} C [21 Proposition 3.13] , which Krylov, Paegelow and Shlykov deduce from [24 Cor. 1.4] by localization. The following proposition proves a generalization to the integral case of [21 Conjecture 4.6] . Note that we add the part A ∘ ⊗ A \mathcal{A}^{\circ}\otimes_{\mathcal{A}} A ∘ ⊗ A K T ( L ( n , r ) ) K_{\mathcal{T}}(\mathcal{L}(n,r)) K T ( L ( n , r ))
Proposition 5.3.
(1) Let R R R q q q Q 1 , … , Q r Q_1,\ldots,Q_r Q 1 , … , Q r q − 1 q-1 q − 1 Z ( H q , n Q ( R ) ) = Z ( H q , n Q ( R ) ) J M Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(R))=Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(R))^{\mathrm{JM}} Z ( H q , n Q ( R )) = Z ( H q , n Q ( R ) ) JM R R R ∣ S n ∣ |\mathfrak{S}_n| ∣ S n ∣
(2) The isomorphisms of [21 Theorem 3.16 and Lemma 4.5] restrict to isomorphisms of Z ( H q , n Q ) Z(\mathcal{H}^{\mathsf{Q}}_{q,n}) Z ( H q , n Q )
A ∘ ⊗ A Z ( H q , n Q ) ≅ Z ( H q , n ∘ ) \mathcal{A}^{\circ}\otimes_{\mathcal{A}}Z(\mathcal{H}^{\mathsf{Q}}_{q,n})\cong Z(\mathcal{H}^{\circ}_{q,n}) A ∘ ⊗ A Z ( H q , n Q ) ≅ Z ( H q , n ∘ ) and
A ∘ ⊗ A K T ( L ( n , r ) ) ≅ Tr ( H q , n ∘ ) . \mathcal{A}^{\circ}\otimes_{\mathcal{A}}K_{\mathcal{T}}(\mathcal{L}(n,r))\cong\operatorname{Tr}(\mathcal{H}^{\circ}_{q,n}). A ∘ ⊗ A K T ( L ( n , r )) ≅ Tr ( H q , n ∘ ) . In particular, Conjecture 5.2 holds.
Proof. Let F F F R R R H q , n Q ( F ) ≅ F ⊗ R H q , n Q ( R ) \mathcal{H}^{\mathsf{Q}}_{q,n}(F)\cong F\otimes_R\mathcal{H}^{\mathsf{Q}}_{q,n}(R) H q , n Q ( F ) ≅ F ⊗ R H q , n Q ( R ) q − 1 q-1 q − 1 R R R q ≠ 1 q\ne1 q = 1 F F F Z ( H q , n Q ( F ) ) = Z ( H q , n Q ( F ) ) J M Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(F))=Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(F))^{\mathrm{JM}} Z ( H q , n Q ( F )) = Z ( H q , n Q ( F ) ) JM
F ⊗ R Z ( H q , n Q ( R ) ) J M → id F ⊗ ι F ⊗ R Z ( H q , n Q ( R ) ) → id F ⊗ π T F ⊗ R Z ( H q , n Q ( R ) ) Z ( H q , n Q ( R ) ) J M ⟶ 0 ↓ p 0 ↓ ι Z ( H q , n Q ( F ) ) J M = Z ( H q , n Q ( F ) ) \begin{array}{ccccc}
F\otimes_R Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(R))^{\mathrm{JM}} & \xrightarrow{\operatorname{id}_F\otimes\iota} & F\otimes_R Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(R)) & \xrightarrow{\operatorname{id}_F\otimes\pi_T} & F\otimes_R \dfrac{Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(R))}{Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(R))^{\mathrm{JM}}} \longrightarrow0 \\
\downarrow p_0 && \downarrow\iota&& \\
Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(F))^{\mathrm{JM}} & = & Z(\mathcal{H}^{\mathsf{Q}}_{q,n}(F))
\end{array} F ⊗ R Z ( H q , n Q ( R ) ) JM ↓ p 0 Z ( H q , n Q ( F ) ) JM id F ⊗ ι = F ⊗ R Z ( H q , n Q ( R )) ↓ ι Z ( H q , n Q ( F )) id F ⊗ π T F ⊗ R Z ( H q , n Q ( R ) ) JM Z ( H q , n Q ( R )) ⟶ 0 where ι : Z ( H q , n Q ( R ) ) J M ↪ Z ( H q , n Q ( R ) ) \iota: Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))^{\mathrm{JM}} \hookrightarrow Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) ι : Z ( H q , n Q ( R ) ) JM ↪ Z ( H q , n Q ( R )) π : Z ( H q , n Q ( R ) ) → Z ( H q , n Q ( R ) ) / Z ( H q , n Q ( R ) ) J M \pi: Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) \to Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))/Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))^{\mathrm{JM}} π : Z ( H q , n Q ( R )) → Z ( H q , n Q ( R )) / Z ( H q , n Q ( R ) ) JM [15 Theorem 1.4(2)] . By definition, p 0 p_0 p 0 id F ⊗ ι \operatorname{id}_{F} \otimes\iota id F ⊗ ι
F ⊗ R Z ( H q , n Q ( R ) ) / Z ( H q , n Q ( R ) ) J M = 0. F \otimes_R Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))/Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))^{\mathrm{JM}} = 0. F ⊗ R Z ( H q , n Q ( R )) / Z ( H q , n Q ( R ) ) JM = 0. By [15 Theorem 1.4(2)] , Z ( H q , n Q ( R ) ) Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) Z ( H q , n Q ( R )) R R R ∣ P n r ∣ |\mathcal{P}^{r}_{n}| ∣ P n r ∣ F = R / m F = R/\mathfrak{m} F = R / m m \mathfrak{m} m R R R Z ( H q , n Q ( R ) ) / Z ( H q , n Q ( R ) ) J M = 0 Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))/Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))^{\mathrm{JM}} = 0 Z ( H q , n Q ( R )) / Z ( H q , n Q ( R ) ) JM = 0 Z ( H q , n Q ( R ) ) J M = Z ( H q , n Q ( R ) ) Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))^{\mathrm{JM}} = Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) Z ( H q , n Q ( R ) ) JM = Z ( H q , n Q ( R )) ∣ P n r ∣ |\mathcal{P}^{r}_{n}| ∣ P n r ∣
For (2), let R = A ∘ R = \mathcal{A}^{\circ} R = A ∘ [27 Lemma 7.1.5] (see also [21 Lemma 3.10] ), K T ( G ( n , r ) ) K_T(\mathcal{G}(n,r)) K T ( G ( n , r )) K T ( L ( n , r ) ) K_T(\mathcal{L}(n,r)) K T ( L ( n , r )) A \mathcal{A} A K T ( G ( n , r ) ; C ) K_T(\mathcal{G}(n,r);\mathbb{C}) K T ( G ( n , r ) ; C ) K T ( L ( n , r ) ; C ) K_T(\mathcal{L}(n,r);\mathbb{C}) K T ( L ( n , r ) ; C ) R ( G L n × T ) R(GL_n \times T) R ( G L n × T ) A \mathcal{A} A Λ i C n \Lambda^i\mathbb{C}^n Λ i C n 1 ≤ i ≤ n 1 \le i \le n 1 ≤ i ≤ n ( Λ n C n ) ∗ (\Lambda^n\mathbb{C}^n)^* ( Λ n C n ) ∗ Lemma 5.1 and [21 Theorem 3.16] , the isomorphism K T ( G ( n , r ) ; C ) ≅ Z ( H q , n Q ( A ∘ ) ) J M K_T(\mathcal{G}(n,r);\mathbb{C}) \cong Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(\mathcal{A}^{\circ}))^{\mathrm{JM}} K T ( G ( n , r ) ; C ) ≅ Z ( H q , n Q ( A ∘ ) ) JM R ⊗ A K T ( G ( n , r ) ) R \otimes_{\mathcal{A}} K_T(\mathcal{G}(n,r)) R ⊗ A K T ( G ( n , r )) R R R Z ( H q , n Q ( A ∘ ) ) Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(\mathcal{A}^{\circ})) Z ( H q , n Q ( A ∘ )) e 1 ( L ) , … , e n ( L ) e_1(L),\ldots,e_n(L) e 1 ( L ) , … , e n ( L ) e n ( L ) − 1 = ( L 1 ⋯ L n ) − 1 e_n(L)^{-1}=(L_1\cdots L_n)^{-1} e n ( L ) − 1 = ( L 1 ⋯ L n ) − 1 Z ( H q , n Q ( R ) ) Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) Z ( H q , n Q ( R )) Z ( H q , n Q ( R ) ) J M Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))^{\mathrm{JM}} Z ( H q , n Q ( R ) ) JM Z ( H q , n Q ( R ) ) Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) Z ( H q , n Q ( R ))
The proof of [27 Lemma 7.1.5] shows that ⟨ − , − ⟩ \langle-, - \rangle ⟨ − , − ⟩ K T ( G ( n , r ) ) K_T(\mathcal{G}(n,r)) K T ( G ( n , r )) K T ( L ( n , r ) ) K_T(\mathcal{L}(n,r)) K T ( L ( n , r )) A \mathcal{A} A τ \tau τ Tr ( H q , n Q ( R ) ) \operatorname{Tr}(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) Tr ( H q , n Q ( R )) R R R [15 Th. 1.4(1)] ), the trace pairing is perfect over R R R R ⊗ A K T ( L ( n , r ) ) ≅ Tr ( H q , n Q ( R ) ) R \otimes_{\mathcal{A}} K_T(\mathcal{L}(n,r)) \cong\operatorname{Tr}(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) R ⊗ A K T ( L ( n , r )) ≅ Tr ( H q , n Q ( R )) Frac ( A C ) \operatorname{Frac}(\mathcal{A}_{\mathbb{C}}) Frac ( A C ) ψ \psi ψ ψ \psi ψ R ⊗ A K T ( L ( n , r ) ) ≅ Tr ( H q , n Q ( R ) ) R \otimes_{\mathcal{A}} K_T(\mathcal{L}(n,r)) \cong\operatorname{Tr}(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) R ⊗ A K T ( L ( n , r )) ≅ Tr ( H q , n Q ( R )) Z ( H q , n Q ( R ) ) Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) Z ( H q , n Q ( R )) ψ \psi ψ [21 Conjecture 1.2, Conjecture 4.6] . □ \square □
Remark 5.4. The hypothesis that q − 1 q-1 q − 1 [15 Th. 1.4(2)] , the equality Z ( H q , n Q ( R ) ) = Z ( H q , n Q ( R ) ) J M Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R)) = Z(\mathcal{H}^{\mathbb{Q}}_{q,n}(R))^{\mathrm{JM}} Z ( H q , n Q ( R )) = Z ( H q , n Q ( R ) ) JM R = A R=\mathcal{A} R = A q = 1 q=1 q = 1 r = 1 r=1 r = 1 n ≥ 2 n\ge2 n ≥ 2 q = Q 1 = 1 q=Q_1=1 q = Q 1 = 1 H q , n Q ( k ) = k S n \mathcal{H}^{\mathbb{Q}}_{q,n}(k)=k\mathfrak{S}_n H q , n Q ( k ) = k S n L 1 = ⋯ = L n = 1 L_1=\cdots=L_n=1 L 1 = ⋯ = L n = 1 k k k Z ( k S n ) Z(k\mathfrak{S}_n) Z ( k S n ) [2 §2] .
Let X X X K 0 v b ( X ) K^{\mathrm{vb}}_0(X) K 0 vb ( X ) G 0 ( X ) G_0(X) G 0 ( X )
K 0 v b ( X ) ⟶ G 0 ( X ) , [ E ] ↦ [ E ] . K^{\mathrm{vb}}_0(X) \longrightarrow G_0(X), \qquad[E] \mapsto[E]. K 0 vb ( X ) ⟶ G 0 ( X ) , [ E ] ↦ [ E ] . By [13 Section 15.1, pp. 281–282] , this is an isomorphism because X X X K K K X X X
K ( X ) = K 0 v b ( X ) ≅ G 0 ( X ) . (12) K(X)=K^{\mathrm{vb}}_0(X)\cong G_0(X). \tag*{(12)} K ( X ) = K 0 vb ( X ) ≅ G 0 ( X ) . (12) The product in K ( X ) K(X) K ( X ) k k k K ( X ; k ) = k ⊗ Z K ( X ) K(X;k)=k\otimes_{\mathbb{Z}}K(X) K ( X ; k ) = k ⊗ Z K ( X )
Now let k k k e e e ζ ∈ k \zeta\in k ζ ∈ k e e e κ = ( κ 1 , … , κ r ) ∈ Z r \boldsymbol{\kappa}=(\kappa_1,\ldots,\kappa_r)\in\mathbb{Z}^r κ = ( κ 1 , … , κ r ) ∈ Z r H ζ , n Λ ( k ) = H q , n Q ( k ) \mathcal{H}^{\Lambda}_{\zeta,n}(k)=\mathcal{H}^{\mathbb{Q}}_{q,n}(k) H ζ , n Λ ( k ) = H q , n Q ( k ) q = ζ q=\zeta q = ζ Q m = ζ κ m Q_m=\zeta^{\kappa_m} Q m = ζ κ m I = Z / e Z I=\mathbb{Z}/e\mathbb{Z} I = Z / e Z c ∈ Z / e Z c\in\mathbb{Z}/e\mathbb{Z} c ∈ Z / e Z ζ c \zeta^c ζ c Λ = ∑ m Λ κ ‾ m \Lambda=\sum_m\Lambda_{\overline{\kappa}_m} Λ = ∑ m Λ κ m κ ‾ m \overline{\kappa}_m κ m κ m \kappa_m κ m e e e α ∈ Q n + \alpha\in Q^+_n α ∈ Q n + H ζ , α Λ ( k ) \mathcal{H}^{\Lambda}_{\zeta,\alpha}(k) H ζ , α Λ ( k ) H ζ , n Λ ( k ) \mathcal{H}^{\Lambda}_{\zeta,n}(k) H ζ , n Λ ( k ) R α Λ ( k ) R^\Lambda_\alpha(k) R α Λ ( k ) Theorem 4.3 , and 1 α 1_\alpha 1 α Λ = ∑ i w i Λ i \Lambda=\sum_i w_i\Lambda_i Λ = ∑ i w i Λ i α = ∑ i v i α i \alpha=\sum_i v_i\alpha_i α = ∑ i v i α i [21 §5.3] and [29 ] , this is ℓ = e \ell=e ℓ = e s m = κ ‾ m s_m=\overline{\kappa}_m s m = κ m Γ \Gamma Γ g g g d = α d=\alpha d = α M d Γ s \mathcal{M}^{\Gamma_s}_d M d Γ s M Λ , α \mathcal{M}_{\Lambda,\alpha} M Λ , α B d B_d B d H ζ , α Λ ( C ) \mathcal{H}^{\Lambda}_{\zeta,\alpha}(\mathbb{C}) H ζ , α Λ ( C )
Proof of Theorem C. Parts (1) and (2) follow from Proposition 5.3 . For (3), first let k = C k=\mathbb{C} k = C [21 Theorem 5.4] , whose hypothesis [21 Conjecture 4.6] holds by Proposition 5.3 , since their proof uses it only through [21 Proposition 4.7] . Since the isomorphism there is left a bit implicit, let us explain it in more detail, following [21 §5.3] . We can consider the tuple g = ( q , Q 1 , … , Q r ) = ( ζ , ζ κ 1 , … , ζ κ r ) g=(q,Q_1,\ldots,Q_r)=(\zeta,\zeta^{\kappa_1},\ldots,\zeta^{\kappa_r}) g = ( q , Q 1 , … , Q r ) = ( ζ , ζ κ 1 , … , ζ κ r ) T \mathcal{T} T G ( n , r ) g \mathcal{G}(n,r)^g G ( n , r ) g w \mathbf{w} w Λ \Lambda Λ [29 Theorem 2.15 & Proposition 2.17] . The dimension vectors that occur are those α ∈ Q n + \alpha\in Q_n^+ α ∈ Q n + M Λ , α ≠ ∅ \mathcal{M}_{\Lambda,\alpha}\ne\varnothing M Λ , α = ∅ K ( G ( n , r ) g ; C ) = ⨁ α K ( M Λ , α ; C ) K(\mathcal{G}(n,r)^g;\mathbb{C})=\bigoplus_\alpha K(\mathcal{M}_{\Lambda,\alpha};\mathbb{C}) K ( G ( n , r ) g ; C ) = ⨁ α K ( M Λ , α ; C ) [21 Proposition 5.3] . Let C g \mathbb{C}_g C g C \mathbb{C} C A \mathcal{A} A q ↦ ζ q\mapsto\zeta q ↦ ζ Q m ↦ ζ κ m Q_m\mapsto\zeta^{\kappa_m} Q m ↦ ζ κ m ζ ≠ 1 \zeta\ne1 ζ = 1 A ∘ \mathcal{A}^{\circ} A ∘ g g g e e e T \mathcal{T} T E E E G ( n , r ) g \mathcal{G}(n,r)^g G ( n , r ) g E ∣ G ( n , r ) g = ⨁ c ∈ Z / e Z E c E|_{\mathcal{G}(n,r)^g}=\bigoplus_{c\in\mathbb{Z}/e\mathbb{Z}}E_c E ∣ G ( n , r ) g = ⨁ c ∈ Z / e Z E c g g g E c E_c E c ζ c \zeta^c ζ c
sp g : C g ⊗ A K T ( G ( n , r ) ) ⟶ K ( G ( n , r ) g ; C ) , [ E ] ⟼ ∑ c ∈ Z / e Z ζ c [ E c ] (13) \operatorname{sp}_g:\mathbb{C}_g\otimes_{\mathcal{A}}K_{\mathcal{T}}(\mathcal{G}(n,r))\longrightarrow K(\mathcal{G}(n,r)^g;\mathbb{C}),\qquad[E]\longmapsto\sum_{c\in\mathbb{Z}/e\mathbb{Z}}\zeta^c[E_c] \tag*{(13)} sp g : C g ⊗ A K T ( G ( n , r )) ⟶ K ( G ( n , r ) g ; C ) , [ E ] ⟼ c ∈ Z / e Z ∑ ζ c [ E c ] (13) is a ring homomorphism; by the localization theorem it is an isomorphism, but we will not need this. Composing it with Proposition 5.3(2) and the base change isomorphism of [15 Theorem 1.4(2)] , we obtain a ring homomorphism
Φ : Z ( H ζ , n Λ ( C ) ) ≅ C g ⊗ A ∘ Z ( H q , n ∘ ) ≅ C g ⊗ A K T ( G ( n , r ) ) → sp g ⨁ α ∈ Q n + K ( M Λ , α ; C ) \Phi:Z(\mathcal{H}^{\Lambda}_{\zeta,n}(\mathbb{C}))\cong\mathbb{C}_g\otimes_{\mathcal{A}^{\circ}}Z(\mathcal{H}^{\circ}_{q,n})\cong\mathbb{C}_g\otimes_{\mathcal{A}}K_{\mathcal{T}}(\mathcal{G}(n,r))\xrightarrow{\operatorname{sp}_g}\bigoplus_{\alpha\in Q_n^+}K(\mathcal{M}_{\Lambda,\alpha};\mathbb{C}) Φ : Z ( H ζ , n Λ ( C )) ≅ C g ⊗ A ∘ Z ( H q , n ∘ ) ≅ C g ⊗ A K T ( G ( n , r )) sp g α ∈ Q n + ⨁ K ( M Λ , α ; C ) with Φ ( e i ( L 1 , … , L n ) ) = sp g [ Λ i V ] \Phi(e_i(L_1,\ldots,L_n))=\operatorname{sp}_g[\Lambda^i\mathcal{V}] Φ ( e i ( L 1 , … , L n )) = sp g [ Λ i V ] V \mathcal{V} V G ( n , r ) \mathcal{G}(n,r) G ( n , r )
On M Λ , α \mathcal{M}_{\Lambda,\alpha} M Λ , α V = ⨁ c V c \mathcal{V}=\bigoplus_c\mathcal{V}_c V = ⨁ c V c V c \mathcal{V}_c V c sp g [ Λ i V ] \operatorname{sp}_g[\Lambda^i\mathcal{V}] sp g [ Λ i V ] i i i ζ c x \zeta^c x ζ c x x x x K K K V c \mathcal{V}_c V c L a e ( i ) = ζ i a ( 1 + y a ) e ( i ) L_a e(i)=\zeta^{i_a}(1+y_a)e(i) L a e ( i ) = ζ i a ( 1 + y a ) e ( i ) b ( u ) = 1 + u b(u)=1+u b ( u ) = 1 + u (9) and (10) , the idempotent 1 α 1_\alpha 1 α z c ; j = ∑ i ∈ I α e j ( L a ∣ i a = c ) e ( i ) z_{c;j}=\sum_{i\in I^\alpha}e_j(L_a\mid i_a=c)e(i) z c ; j = ∑ i ∈ I α e j ( L a ∣ i a = c ) e ( i ) L 1 , … , L n L_1,\ldots,L_n L 1 , … , L n x − 1 x-1 x − 1 y a y_a y a ζ c x \zeta^c x ζ c x M Λ , α \mathcal{M}_{\Lambda,\alpha} M Λ , α ζ c j [ Λ j V c ] \zeta^{cj}[\Lambda^j\mathcal{V}_c] ζ c j [ Λ j V c ] Φ \Phi Φ Z ( H ζ , α Λ ( C ) ) Z(\mathcal{H}^{\Lambda}_{\zeta,\alpha}(\mathbb{C})) Z ( H ζ , α Λ ( C )) K ( M Λ , α ; C ) K(\mathcal{M}_{\Lambda,\alpha};\mathbb{C}) K ( M Λ , α ; C ) [29 Proposition 5.3] ), sending ζ − c j z c ; j = ∑ i e j ( 1 + y a ∣ i a = c ) e ( i ) \zeta^{-cj}z_{c;j}=\sum_i e_j(1+y_a\mid i_a=c)e(i) ζ − c j z c ; j = ∑ i e j ( 1 + y a ∣ i a = c ) e ( i ) [ Λ j V c ] [\Lambda^j\mathcal{V}_c] [ Λ j V c ] e c ; j e_{c;j} e c ; j e j ( x − 1 ) e_j(x-1) e j ( x − 1 ) x x x V c \mathcal{V}_c V c [ Λ j V c ] [\Lambda^j\mathcal{V}_c] [ Λ j V c ] Λ v c V c \Lambda^{v_c}\mathcal{V}_c Λ v c V c K ( M Λ , α ) K(\mathcal{M}_{\Lambda,\alpha}) K ( M Λ , α ) Φ \Phi Φ Φ : Z ( H ζ , α Λ ( C ) ) → K ( M Λ , α ; C ) \Phi:Z(\mathcal{H}^{\Lambda}_{\zeta,\alpha}(\mathbb{C}))\to K(\mathcal{M}_{\Lambda,\alpha};\mathbb{C}) Φ : Z ( H ζ , α Λ ( C )) → K ( M Λ , α ; C ) ∣ P α r ∣ |\mathcal{P}^{r}_{\alpha}| ∣ P α r ∣ Φ \Phi Φ Z ( H ζ , α Λ ( C ) ) ≅ K ( M Λ , α ; C ) Z(\mathcal{H}^{\Lambda}_{\zeta,\alpha}(\mathbb{C}))\cong K(\mathcal{M}_{\Lambda,\alpha};\mathbb{C}) Z ( H ζ , α Λ ( C )) ≅ K ( M Λ , α ; C ) k = C \mathbb{k}=\mathbb{C} k = C
Now let k \mathbb{k} k ζ C = exp ( 2 π i / e ) \zeta_{\mathbb{C}}=\exp(2\pi i/e) ζ C = exp ( 2 π i / e ) O = Z [ ζ C , 1 / e ] ⊂ C \mathcal{O}=\mathbb{Z}[\zeta_{\mathbb{C}},1/e]\subset\mathbb{C} O = Z [ ζ C , 1/ e ] ⊂ C Q ( ζ C ) w i t h \mathbb{Q}(\zeta_{\mathbb{C}}) with Q ( ζ C ) w i t h k \mathbb{k} k e e e O → k \mathcal{O}\to\mathbb{k} O → k ζ C ↦ ζ \zeta_{\mathbb{C}}\mapsto\zeta ζ C ↦ ζ
The element ζ C − 1 \zeta_{\mathbb{C}}-1 ζ C − 1 O \mathcal{O} O ∏ k = 1 e − 1 ( 1 − ζ C k ) = e \prod_{k=1}^{e-1}(1-\zeta_{\mathbb{C}}^k)=e ∏ k = 1 e − 1 ( 1 − ζ C k ) = e R = O R=\mathcal{O} R = O q = ζ C q=\zeta_{\mathbb{C}} q = ζ C Q m = ζ C κ m Q_m=\zeta_{\mathbb{C}}^{\kappa_m} Q m = ζ C κ m Z ( H ζ , n Λ ( O ) ) Z(\mathcal{H}^{\Lambda}_{\zeta,n}(\mathcal{O})) Z ( H ζ , n Λ ( O )) O \mathcal{O} O [15 Theorem 1.4(2)] its base changes to C \mathbb{C} C k \mathbb{k} k Z ( H ζ , n Λ ( C ) ) Z(\mathcal{H}^{\Lambda}_{\zeta,n}(\mathbb{C})) Z ( H ζ , n Λ ( C )) Z ( H ζ , n Λ ( k ) ) Z(\mathcal{H}^{\Lambda}_{\zeta,n}(\mathbb{k})) Z ( H ζ , n Λ ( k ))
Furthermore, the idempotent 1 α 1_\alpha 1 α z c ; j z_{c;j} z c ; j O \mathcal{O} O L a L_a L a O \mathcal{O} O ζ c − ζ c ′ \zeta^c-\zeta^{c'} ζ c − ζ c ′ c ≠ c ′ c\ne c' c = c ′ Z / e Z \mathbb{Z}/e\mathbb{Z} Z / e Z O \mathcal{O} O 1 α Z ( H α Λ ( O ) ) 1_\alpha Z(\mathcal{H}^{\Lambda}_{\alpha}(\mathcal{O})) 1 α Z ( H α Λ ( O )) Z ( H α Λ ( C ) ) Z(\mathcal{H}^{\Lambda}_{\alpha}(\mathbb{C})) Z ( H α Λ ( C )) k \mathbb{k} k Z ( H α Λ ( k ) ) Z(\mathcal{H}^{\Lambda}_{\alpha}(\mathbb{k})) Z ( H α Λ ( k ))
The isomorphism Φ \Phi Φ 1 α Z ( H α Λ ( O ) ) 1_\alpha Z(\mathcal{H}^{\Lambda}_{\alpha}(\mathcal{O})) 1 α Z ( H α Λ ( O )) K ( M Λ , α ; O ) K(\mathcal{M}_{\Lambda,\alpha};\mathcal{O}) K ( M Λ , α ; O ) Z ( H α Λ ( O ) ) Z(\mathcal{H}^{\Lambda}_{\alpha}(\mathcal{O})) Z ( H α Λ ( O )) e 1 ( L ) , … , e n ( L ) e_1(L),\ldots,e_n(L) e 1 ( L ) , … , e n ( L ) sp g [ Λ i V ] \operatorname{sp}_g[\Lambda^i\mathcal{V}] sp g [ Λ i V ] O \mathcal{O} O [ Λ j V c ] = Φ ( ζ − c j z c ; j ) [\Lambda^j\mathcal{V}_c]=\Phi(\zeta^{-cj}z_{c;j}) [ Λ j V c ] = Φ ( ζ − c j z c ; j ) [ Λ v c V c ] [\Lambda^{v_c}\mathcal{V}_c] [ Λ v c V c ] z c ; v c z_{c;v_c} z c ; v c 1 α Z ( H α Λ ( O ) ) 1_\alpha Z(\mathcal{H}^{\Lambda}_{\alpha}(\mathcal{O})) 1 α Z ( H α Λ ( O )) L a L_a L a L a − 1 L_a^{-1} L a − 1 K ( M Λ , α ; O ) K(\mathcal{M}_{\Lambda,\alpha};\mathcal{O}) K ( M Λ , α ; O ) C \mathbb{C} C 1 α Z ( H α Λ ( O ) ) ≅ K ( M Λ , α ; O ) 1_\alpha Z(\mathcal{H}^{\Lambda}_{\alpha}(\mathcal{O}))\cong K(\mathcal{M}_{\Lambda,\alpha};\mathcal{O}) 1 α Z ( H α Λ ( O )) ≅ K ( M Λ , α ; O ) k \mathbb{k} k O \mathcal{O} O Z ( H α Λ ( k ) ) ≅ K ( M Λ , α ; k ) Z(\mathcal{H}^{\Lambda}_{\alpha}(\mathbb{k}))\cong K(\mathcal{M}_{\Lambda,\alpha};\mathbb{k}) Z ( H α Λ ( k )) ≅ K ( M Λ , α ; k )
Acknowledgements
Acknowledgements. We thank Vasily Krylov, Raphaël Paegelow and Pavel Shlykov for helpful discussions on their work [21 ] . JH was supported by the National Natural Science Foundation of China (No. 12431002). AM was partially supported by the Australian Research Council DP240101809. BW was supported by the NSERC through a Discovery Grant. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. We acknowledge the use of AI in proofreading this manuscript.
References
Key Laboratory of Algebraic Lie Theory and Analysis of Ministry of Education, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China
Email address: junhu404@bit.edu.cn
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Email address: andrew.mathas@sydney.edu.au
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Email address: leishi202406@163.com
Department of Pure Mathematics, University of Waterloo & Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
Email address: ben.webster@uwaterloo.ca
