The modular branching rule for affine Hecke algebras of type A

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The modular branching rule for affine Hecke algebras of type A

Susumu Ariki, Nicolas Jacon, Cédric Lecouvey

To cite this version:

Susumu Ariki, Nicolas Jacon, Cédric Lecouvey. The modular branching rule for affine Hecke algebras of type A. Advances in Mathematics, Elsevier, 2011, 228 (1), pp.481-526. �hal-00315397v3�

THE MODULAR BRANCHING RULE FOR AFFINE HECKE ALGEBRAS OF TYPE A

SUSUMU ARIKI, NICOLAS JACON AND C´EDRIC LECOUVEY

Abstract. For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazi- rani.

1. Introduction

In [6], Ginzburg explains his geometric construction of simple modules over (extended) affine Hecke algebras H_n defined over C. In this paper, we consider the affine Hecke algebra of type A whose parameter is a root of unity. Then, the simple modules are labelled by aperiodic multisegments.

On the other hand, Dipper, James and Mathas’ Specht module theory gives us a combinatorial construction of simple modules of cyclotomic Hecke algebras, and they exhaust all the simple modules of the affine Hecke algebra.

The simple modules are labelled by Kleshchev multipartitions.

If one wants to compute something about simples, the combinatorially defined simple modules often have more advantage than the geometrically defined simple modules, and we may work over any algebraically closed field. On the other hand, the geometrically defined simple modules are very useful in several circumstances. Hence, explicit description of the module correspondence between the two constructions is desirable.

We provide this explicit description of the module correspondence in this article. Note that both the set of aperiodic multisegments and the set of Kleshchev multipartitions have structure of Kashiwara crystals. Then, we show that the crystal embedding gives the module correspondence. We also describe the crystal embedding explicitly.

Closely related to this result is the modular branching rule. One may prove the result on the module correspondence by using this, which is our first proof, or one may prove the modular branching rule by first establishing the result on the module correspondence, which is our second proof. Note that we mean here the modular branching rule in the original sense as we explain in the next paragraph. Some authors use the terminology in weaker sense, which does not imply the module correspondence.

Date: Revised Mar.19, 2010.

1

LetL_ψ be the simple module labelled by a multisegmentψ, whose precise meaning will be explained in section 4. Themodular branching ruleis a rule to describe Soc(i- Res^H_Hⁿ

n−1(L_ψ)), or equivalently Top(i- Res^H_Hⁿ

n−1(L_ψ)). We show that

Soc(i- Res^H_Hⁿ

n−1(L_ψ)) =L_e˜iψ,

where ˜eiis the Kashiwara operator. We give a geometric proof of this rule in the framework of Lusztig and Ginzburg’s theory. This gives the first proof.

On the other hand, if one uses results in [2] and [3], both become easier, and this is the second proof.

Recall that the main result of [27] is the modular branching rule when the parameter of the affine Hecke algebra is not a root of unity. Hence our result generalizes [27, Theorem 3.1]. In [1] and [5] it was proved that affine sle controls the modular representation theory of cyclotomic Hecke algebras. Later¹, Grojnowski gave another proof [8, Theorem 14.2, 14.3] for several main results in [5]. We note here that he writes in [8, 14.1] that his IMRN paper proved the part concerning the canonical basis in [1], but his announcement in 1995 was that he had some computation of Kazhdan- Lusztig polynomials which he was able to deduce from the IMRN paper:

the use of i-induction and i-restriction functors, integrable modules over g(A⁽¹⁾_e−1), and Lusztig’s aperiodicity were absent in the assertion.

The idea of his proof in [8] came from Leclerc’s observation that Kleshchev and Brundan’s work on the modular branching rule of the symmetric group and the Hecke algebra of typeAmay be understood in crystal language. The proof is interesting, but the modular branching rule in our sense is not proved in [8] and it is natural to ask whether the crystal he used coincides with the one used in [1] and [5]. It was settled affirmatively in [3], but it still used several results from [8]. Here in this paper, the modular branching rule for the affine Hecke algebra, which is a stronger statement than the statement in [8] that the modular branching gives a crystal which is isomorphic to B(∞), is proved in a direct manner. It still uses the multiplicity one result from [9], but it replaces [8].

The paper is organized as follows. In section 2, we review basic facts on the crystal B(∞) of type A⁽¹⁾_e−1. In section 3, we prepare for a geometric proof of the modular branching rule of the affine Hecke algebra. In section 4, we give the geometric proof of the modular branching rule. In section 5, we introduce crystals of deformed Fock spaces and state results to compute crystal isomorphisms among them. In section 6, we prove a lemma on the module correspondence of simple modules in various labellings and give a combinatorial proof of the modular branching rule in the framework of Fock space theory for cyclotomic Hecke algebras.

Acknowledgements. Part of this work was done while the authors were

1Compare the submission date of [5] with [8].

visiting the MSRI in Berkeley in 2008. The authors wish to thank the in- stitute for its hospitality and the organizers of the two programs for their invitation. The second author is also grateful to Hyohe Miyachi for fruitful discussions there. The second author is supported by the “Agence Nationale de la Recherche” (project JCO7-192339).

2. Preliminaries

Lete≥2 be a fixed integer, g the Kac-Moody Lie algebra of typeA⁽¹⁾_e−1. We denote by U_v⁻ the negative part of the quantum affine algebra U_v(g), which is generated by the Chevalley generators f_i, wherei∈Z/eZ, subject to the quantum Serre relations. In this section, we review basic facts onU_v⁻ and its crystal. We denote the simple roots by αi, and the simple coroots by α^∨_i , fori∈Z/eZ.

2.1. The crystal B(∞). Let us introduce the Kashiwara operator ˜f_i, for i∈Z/eZ, on U_v⁻. Letei,i∈Z/eZ, be Chevalley generators of the positive part ofU_v(g) andt_i =v^α∨i. The following two lemmas are due to Kashiwara.

Lemma 2.1. For each u ∈ U_v⁻, there exist unique u^′ and u^′′ in U_v⁻ such that we have

e_iu−ue_i= t_iu^′−t⁻¹_i u^′′

v−v⁻¹ .

We define an operator e^′_i on U_v⁻ by e^′_iu = u^′′, for u ∈ U_v⁻. The algebra generated by{fi}i∈Z/eZ and{e^′_i}i∈Z/eZ is called theKashiwara algebra. Let f_i⁽ⁿ⁾ be then^th divided power off_i.

Lemma 2.2. Let P ∈U_v⁻. For each i∈ Z/eZ, there exists u_n in U_v⁻, for n∈Z_≥0, such that e^′_iu_n= 0, for alln, and P =P

n∈Z_≥0f_i⁽ⁿ⁾u_n. We define ˜e_iP = P

n∈Z_≥1f_i⁽ⁿ⁻¹⁾u_n and ˜f_iP = P

n∈Z_≥0f_i⁽ⁿ⁺¹⁾u_n. They are well-defined. LetR be the subring ofC(v) consisting of elements which are regular at v= 0. Then, we define

L(∞) = X

N∈Z_≥0

X

(i1,...,iN)∈(Z/eZ)^N

Rf˜_i1· · ·f˜_iN1 and

B(∞) =

∪N∈Z_≥0∪_{(i1,...,iN)∈(}Z/eZ)^N f˜_i1· · ·f˜_iN1 +vL(∞)

\ {0}.

B(∞) is a basis of the C-vector space L(∞)/vL(∞). U_v⁻ admits a root space decomposition U_v⁻ = ⊕α∈Q+(U_v⁻)_−α, where Q₊ = P

i∈Z/eZZ_≥0α_i, and it follows that

B(∞) = G

α∈Q+

B(∞)−α.

We define wt(b) =−α if b∈B(∞)−α. Then, by defining

ǫ_i(b) = max{k∈Z_≥0 |e˜^k_ib6= 0} and ϕ_i(b) =ǫ_i(b) + wt(b)(α^∨_i ),

forb∈B(∞), (B(∞),wt, ǫ_i, ϕ_i,e˜_i,f˜_i) is ag-crystal in the sense of Kashiwara [14, p.48].

We define the bar operation onU_v⁻ by ¯v =v⁻¹ and ¯f_i =f_i. Lusztig and Kashiwara independently constructed the canonical basis/the global basis

{Gv(b)|b∈B(∞)}

of U_v⁻, which is characterized by the property that G_v(b) =G_v(b), G_v(b) +vL(∞) =b.

Example 2.3. Let e = 3. Then, e₂ and f₁ commute so that e^′₂f₁ = 0 and ˜f₂f₁ = f₂f₁ follows. Similarly, ˜f₁f₂ = f₁f₂. Thus, {f1f₂, f₂f₁} is the canonical basis of (U_v⁻)−α₁−α₂. For the null root δ = α₀ +α₁ +α₂, {f0f₁f₂, f₂f₁f₀, f₀f₂f₁, f₁f₀f₂} is the canonical basis of (U_v⁻)_−δ. Of course, more complex linear combination of monomials inf_iappear in the canonical basis of other (U_v⁻)_−α.

2.2. Hall algebras. The crystalB(∞) has a concrete description. Let Γ be the cyclic quiver of length e. This is an oriented graph with vertices Z/eZ and edges {(i, i+ 1), i∈Z/eZ}. Let V =⊕_i∈Z/eZVi be a finite dimensional Z/eZ-graded vector space, and define

E_V = M

i∈Z/eZ

HomC(V_i, V_i+1)⊆EndC(V).

An elementX ∈EV is called arepresentation of Γ onV. The vector dimV = (dimVi)_i∈Z/eZ

is called thedimension vector of the representation.

If V runs through all finite dimensional Z/eZ-graded vector spaces, we obtain the category of representations of Γ. It is the same as the category of finite dimensionalCΓ-modules, where CΓ is the path algebra of Γ. IfX is nilpotent as an endomorphism ofV, we say that the representation X, or the correspondingCΓ-module, isnilpotent. We denote by NV the subset of nilpotent representations in E_V. LetG_V =Q

i∈Z/eZGL(V_i). It acts on E_V and NV by conjugation and two representations are equivalent if and only if they are in the sameG_V-orbit.

For each i∈Z/eZ, let V =V_i =Cand X = 0. Then it defines a simple CΓ-module. We denote it by S_i. They are nilpotent representations.

Example 2.4. Let G_n = GL_n(C) and suppose that s ∈ G_n has order e.

Letζ be a primitivee^th root of unity,V =Cⁿ, and letV_i be the eigenspace of s for the eigenvalue ζⁱ. If X ∈ EndC(V) is such that sXs⁻¹ =ζX then XV_i ⊆V_i+1. Thus, X defines a representation of Γ on V. Note that G_V is the centralizer groupGn(s) in this case.

By linear algebra, the isomorphism classes of nilpotent representations are labelled by (Z/eZ-valued) multisegments.

Definition 2.5. Let l ∈ Z_>0 and i ∈ Z/eZ. The segment of length l and headiis the sequence of consecutive residues [i, i+ 1, ..., i+l−1]. We denote it by [i;l). Similarly, The segment of length l and tail i is the sequence of consecutive residues [i−l+ 1, ..., i−1, i]. We denote it by (l;i]. We say that [i;l) has aremovable i-node and [i+ 1;l) has an addable i-node.

A collection of segments is called a multisegment. If the collection is the empty set, we call it the empty multisegment.

Each [i;l) defines an indecomposable nilpotent CΓ-module C[i;ℓ), which is characterized by the property that

C[i;l) is a uniserial module and Top(C[i;l)) =S_i.

Hence, a complete set of isomorphism classes of nilpotent representations is given by the modules

M_ψ = M

i∈Z/eZ,l∈Z_>0

C[i, l)^⊕m[i;l), which is labelled by the multisegment

ψ={[i;l)^⊕m[i;l)}_i∈Z/eZ,l∈Z_>0. We denote the correspondingG_V-orbit in NV byOψ.

Now, we introduce the Hall polynomials. Let F_q be a finite field, and consider F_qΓ-modules. Then, they are classified by multisegments again.

LetV,T and W beZ/eZ-graded vector spaces overF_q such that dimV = dimT+ dimW.

Let ϕ₁, ϕ₂ and ψ be multisegments such that Oϕ1 ⊆ NT, Oϕ2 ⊆ NW and Oψ ⊆ NV. If the number of submodules U of M_ψ that satisfies U ≃ M_ϕ2 and M_ψ/U ≃M_ϕ1 is polynomial in q = card(F_q), then this polynomial is called the Hall polynomial and we denote it by Fϕ^ψ1,ϕ2(q). The existence of Hall polynomials in our case was proved by Jin Yun Guo [10, Theorem 2.7].

Foraand b inZ^e we define a bilinear form mby m(a, b) = X

i∈Z/eZ

(a_ib_i+1+a_ib_i).

We remark that this is not the Euler form used by Ringel to define his (twisted) Hall algebra, but the one used by Lusztig, which comes from the difference of dimensions of the fibers of two fiber bundles which appear in his geometric definition of the product, namely in the definition of the induction functor. In his theory, the Euler form appears in the definition of coproduct, namely in the definition of the restriction functor.

Now, Lusztig’s version of the Hall algebra associated to Γ is the C(v)- algebra with basis {uψ |ψ is a multisegment}and product is given by

uϕ1uϕ2 =v^m(dimT,dimW)X

ψ

F_ϕ^ψ_1,ϕ2(v⁻²)u_ψ.

Note that [i; 1) is the multisegment which labels the simple module S_i, for i∈Z/eZ. Then the C(v)-subalgebra generated by these u_[i;1) is called the composition algebra, and we may and do identify it with U_v⁻ by u_[i;1)7→f_i. For the proof, see [23, Theorem 1.20].

Definition 2.6. For each multisegmentψ, we define E_ψ =v^dimOψu_ψ. The set{Eψ |ψ is a multisegment.}is called the PBW basis of the Hall algebra.

Example 2.7. Let e= 3. Then we have

f₁f₂ =E_{[1;2)}+vE{[1;1),[2;1)}, f₂f₁ =E{[1;1),[2;1)}. Similarly, we have

f₀f₁f₂ =E_{[0;3)}+vE{[1;2),[0;1)}+vE{[0;2),[2;1)}+v²E{[0;1),[1;1),[2;1)}, f₂f₁f₀ =E{[2;2),[1;1)}+vE{[0;1),[1;1),[2;1)},

f0f2f1 =E{[0;2),[2;1)}+vE{[0;1),[1;1),[2;1)}, f₁f₀f₂ =E{[1;2),[0;1)}+vE{[0;1),[1;1),[2;1)}.

Note that E{[0;1),[1;1),[2;1)} does not appear with coefficient 1. This is general phenomenon and we need aperiodicity to describe it.

Definition 2.8. A multisegment ψ is aperiodic if, for everyl∈Z_>0, there exists some i ∈ Z/eZ such that the segment of length l and head i does not appear in ψ. Equivalently, a multisegment ψ is aperiodic if, for each l ∈Z_>0, there exists some i∈Z/eZ such that the segment of length l and tail idoes not appear inψ.

The notion of aperiodicity and the following theorem are due to Lusztig.

See [19, 15.3] and [20, Theorem 5.9].

Theorem 2.9. For each b ∈B(∞), the canonical basis element G_v(b) has the form

G_v(b) =E_ψ+ X

ψ^′6=ψ

c_ψ,ψ^′(v)E_ψ^′,

for a unique aperiodic multisegment ψ, such that c_ψ,ψ^′(v)∈C(v) is regular at v= 0 and c_ψ,ψ^′(0) = 0.

Hence, we may label elements of B(∞) by aperiodic multisegments. We identify B(∞) with the set of aperiodic multisegments. Then, we denote the canonical basis by G_v(ψ), for multisegments ψ, hereafter.

Leclerc, Thibon and Vasserot described the crystal structure on the set of aperiodic multisegmentsB(∞) in [18, Theorem 4.1], by using a result by Reineke.

Letψ be a multisegment. Let ψ≥l be the multisegment obtained from ψ by deleting multisegments of length less than l, for l ∈ Z_>0. Let m_[i;l) be the multiplicity of [i;l) inψ. Then, for i∈Z/eZ, we consider

S_l,i=X

k≥l

(m_[i+1;k)−m_[i;k)),

that is, the number of addablei-nodes ofψ_≥lminus the number of removable i-nodes of ψ_≥l. Let ℓ₀ < ℓ₁ < · · · be those l that attain min_l>0S_l,i. The following is the description of the crystal structure given by Leclerc, Thibon and Vasserot.

Theorem 2.10. Let ψ be a multisegment, i∈Z/eZ and let ℓ₀ be as above.

Then, f˜_iψ=ψ_ℓ0,i, where ψ_ℓ0,i is obtained from ψ by adding [i; 1) if ℓ₀ = 1, and by replacing [i+ 1;ℓ₀−1) with[i;ℓ₀) if ℓ₀>1.

2.3. An anti-automorphism of U_v⁻. As the identification of the affine Hecke algebra with the convolution algebra K^Gn×C×(Z_n), which will be explained in the next section, is not canonical, we go back and forth between two identifications. For this reason, we need another labelling by aperiodic multisegments.

LetV =⊕i∈Z/eZV_ibe a graded vector space as before, and define its dual graded vector space byV^∗ =⊕i∈Z/eZV_i^∗ where V_i^∗ = HomC(V_−i,C). Then, by sendingX∈EV to its transpose, we have a linear isomorphism

ρ:E_V ≃E_V^∗ =⊕_i∈Z/eZHomC(V_i^∗, V_i+1^∗ ).

Using the standard basis of E_V and its dual basis in E_V^∗, we identify the underlying spaces EV and EV^∗. Note that theGV-action on thisEV is the conjugation by the transpose inverse ofg∈G_V, while theG_V-action on the originalEV is the conjugation byg∈GV. Then,ρis an isomorphism of two G_V-varietiesE_V so that theG_V-orbitOψ in the originalE_V corresponds to theG_V-orbitO_ρ(ψ)in the newE_V, whereρ(ψ) is defined byρ([i;l)) = (l;−i].

Thus, we have a linear isomorphism of the Hall algebras on both sides, which we also denote byρ, such that

ρ(E_ψ) =E_ρ(ψ) and ρ(G_v(ψ)) =G_v(ρ(ψ)) if ψ is aperiodic.

That is, this gives a relabelling of the PBW basis and the canonical basis.

However, if we take the algebra structure into account, ρ induces the anti- automorphism of U_v⁻ given byfi 7→ f−i, which is clear from the definition of the multiplication of the Hall algebra. In particular, the crystal structure on the set of aperiodic multisegments is changed in this new labelling, and the Kashiwara operators ˜e_i and ˜f_i correspond to the Kashiwara operators

˜

e_−i and ˜f_−i in this new crystal structure. In the new crystal structure, we change the definition of addable and removable i-nodes as follows.

Definition 2.11. We say that (l;i] has aremovable i-node and (l;i−1] has an addable i-node.

We considerS_l,i=P

k≥l(m_(k;i−1]−m_(k;i]), that is, the number of addable i-nodes of ψ_≥l minus the number of removable i-nodes of ψ_≥l in the new definition of removable and addable i-nodes. Let ℓ₀ < ℓ₁ < · · · be those l that attain min_l>0S_l,i. Then, the crystal structure in the new labelling is given as follows. In fact, this version is stated in [18].

Theorem 2.12. Let ψ be a multisegment, i∈Z/eZ and let ℓ₀ be as above.

Then, f˜_iψ=ψ_ℓ0,i, where ψ_ℓ0,i is obtained from ψ by adding (1;i] if ℓ₀ = 1, and by replacing (ℓ₀−1;i−1] with(ℓ₀;i] if ℓ₀>1.

To compute ˜eiψ, for a multisegment ψ, we consider the same S_l,i. If min_l>0S_l,i = 0, then ˜e_iψ = 0. Otherwise, let ℓ₀ be the maximal l that attains min_l>0S_l,i. Then, ˜e_iψ is obtained from ψ by replacing (ℓ₀;i] with (ℓ₀−1;i−1].

We use the crystal structure on the set of aperiodic multisegments in Theorem 2.10 when we choose the identification of R(G_n ×C^×)-algebras H_n≃K^Gn×C×(Z_n) following Lusztig [22], while we use that in Theorem 2.12 when we choose the identification Hn ≃ K^Gn×C×(Zn) following Ginzburg [6]. We note that the second crystal structure is the star crystal structure of the first.

3. Affine Hecke algebras

Let H_n be the extended affine Hecke algebra associated with G_n. It is the C[q^±1]-algebra generated byT_i, for 1≤i < n, andX_i^±1, for 1≤i≤n, subject to the relations

(T_i−q)(T_i+ 1) = 0, q⁻¹T_iX_iT_i=X_i+1, etc.

In this section, we recall the geometric realization of affine Hecke algebras by Lusztig and Ginzburg, and of specialized affine Hecke algebras by Ginzburg.

3.1. Varieties. LetG_n=GL_n(C) as before, andB_nthe Borel subgroup of upper triangular matrices. We denote the unipotent radical ofB_nbyU_n, and the maximal torus of diagonal matrices byTn. WriteCⁿ=Ce1⊕ · · · ⊕Cen

and let Fℓn be the flag variety, which consists of increasing subspaces F = (F_i)_0≤i≤n in Cⁿ such that dimF_i = i, for all i. We consider the diagonal G_n-action on Fℓ_n× Fℓ_n. Then, G_n-orbits in Fℓ_n× Fℓ_n are in bijection withB_n-orbits in Fℓn and if we denote

{(F, F^′)∈ Fℓ_n× Fℓ_n |dim(F_i∩F_j^′) =♯{k|1≤k≤i, 1≤w(k)≤j}}

by O_n(w), for w∈S_n, they give a complete set ofG_n-orbits, and a pair of flags (F, F^′) belongs toOn(w) if and only if

dim F_i∩F_j^′

F_i−1∩F_j^′+F_i∩F_j−1^′ =

(1 (j=w(i)) 0 (otherwise).

We denote by Nn the set of nilpotent elements inM at_n(C) and write Y_n={(X, F) ∈ Nn× Fℓn|XF_i⊆Fi−1} ≃T^∗Fℓ_n. Then the Steinberg variety is defined by

Z_n=Y_n×NnY_n

={(X, F, F^′)∈ Nn× Fℓn× Fℓ_n|XF_i ⊆F_i−1, XF_i^′ ⊆F_i−1^′ }.

Z_n is aG_n×C^×-variety by the action

(g, c)(X, F, F^′) = (c⁻¹gXg⁻¹, gF, gF^′), for (g, c)∈G_n×C^× and (X, F, F^′)∈Z_n.

We consider the complexified K-group of the abelian category ofG_n×C^×- equivariant coherent sheaves on Z_n. Using the closed embedding Z_n ⊆ Y_n×Y_n, we have the convolution algebraK^Gn×C×(Z_n). Z_n has a partition Zn=⊔w∈SnZn(w), where

Z_n(w) ={(X, F, F^′)∈Z_n|(F, F^′)∈O_n(w)}.

We have dimZn(w) =n(n−1) andZn(w) is a (ⁿ⁽ⁿ⁻¹⁾₂ −ℓ(w))-dimensional vector bundle over O_n(w). Then, {Zn(w)}w∈Sn is the set of the irreducible components of Z_n. Define

Zn−1,n={(X, F, F^′)∈Z_n|Fn−1 =F_n−1^′ }.

The condition F_n−1 = F_n−1^′ is equivalent to (F, F^′) ∈ ⊔w∈Sn−1O_n(w), so that we haveZn−1,n =⊔w∈Sn−1Z_n(w).

Similarly, (F, F^′)∈On(e)⊔On(si) =On(si) if and only ifFj =F_j^′, for all j6=i, and

Zn(si) ={(X, F, F^′)∈Zn|Fj =F_j^′, for all j6=i, XFi+1 ⊆Fi−1}.

The pushforward ofO_Z

n(si) with respect to the closed embeddingZ_n(s_i)⊆ Z_n is also denoted by O_Z

n(si) by abuse of notation. We denote b_i = [O_Zn(s

i)]∈K^Gn×C×(Z_n).

LetQ_i,i+1 be the parabolic subgroup ofG_nwhich corresponds tos_i,n_i,i+1 the nilradical of its Lie algebra. Then

Z_n(s_i) = (G_n×C^×)×_Qi,i+1×C^×(n_i,i+1×P¹×P¹)

is a vector bundle over On(si) = (Gn×C^×)×_Qi,i+1×C^×(P¹×P¹). Then we define as follows.

Definition 3.1. The line bundleL_i onZ_n(s_i) is the pullback of (G_n×C^×)×_Qi,i+1×C^×(OP¹(−1)⊗ OP¹(−1)) on O_n(s_i).

For λ∈Zǫ1⊕ · · · ⊕Zǫn= Hom(Tn,C^×), let C_λ be the Bn×C^×-module associated withλand define the associated line bundle L_λ onFℓ_n by

L_λ = (Gn×C^×)×_Bn×C^×C_λ.

When we consider λ as a character of T_n, we denote it by e^λ. Then, we identifyK^Gn×C×(Fℓ_n) =R(T_n×C^×) via L_λ 7→e^λ as usual.

Let us denote π_n : Y_n → Fℓn and δ_n : Z_n(e) ⊆ Z_n. We consider the diagram

Fℓ_n←−^πn Y_n≃Z_n(e)−→^δn Z_n

and we denote

θ_λ = [δ_n∗π_n^∗L_−λ]∈K^Gn×C×(Z_n).

Definition 3.2. We defineTi = [Li] +q, for 1 ≤i < n, and Xi =θǫi, for 1≤i≤n.

We have θ_λ=Qn

i=1X_i^λi, forλ=Pn

i=1λ_iǫ_i. Using the exact sequence 0→ OP¹(−1)⊗ OP¹(−1)→ OP¹ ⊗ OP¹ → O_∆P¹ →0

where ∆P¹ ⊆P¹×P¹ is the diagonal, we know that [Li] =b_i−(1−qθ_αi).

Then, T_i, for 1 ≤ i < n, and X_i^±1, for 1 ≤ i ≤ n, satisfy the defining relations ofH_n. In particular, we have the Bernstein relation

Tiθ_λ =θ_siλTi+ (1−q)θ_λ−θ_siλ θ−αi−1,

where α_i = −ǫ_i+ǫ_i+1. This follows from the next theorem. The theorem was found by Lusztig and the action of T_i is called the Demazure-Lusztig operator.

Theorem 3.3. Through the Thom isomorphism, we identify K^Gn×C×(Y_n) with

K^Gn×C×(Fℓ_n) =R(T_n×C^×).

Then the convolution action of K^Gn×C×(Z_n) on K^Gn×C×(Y_n) is given by Tif = f −sif

e^αi −1 −qf −e^αisif

e^αi−1 , Xif =e^−ǫif.

It is well-known that this is a faithful representation ofH_n. Note that we have chosen the isomorphismH_n≃K^Gn×C×(Z_n) to have the same formulas as [6, Theorem 7.2.16, Proposition 7.6.38]. When we follow [22], we define

θ_λ= [δ_n∗π_n^∗L_λ] and T_i=−[Li]−1.

Then, the formulas for the convolution action onR(T_n×C^×) change to those in [22, p.335]. The two identifications of H_n ≃K^Gn×C×(Z_n) are related by the involution σ defined by

Ti 7→ −qT_i⁻¹, Xi 7→X_i⁻¹.

In the rest of this section, we follow the identification in [22].

The center Z(H_n) of H_n is the C[q^±1]-subalgebra consisting of all the symmetric Laurent polynomials in X1, . . . , Xn. Thus, we identify Z(Hn) withR(G_n×C^×). We also identifyC[q^±1][X₁^±1, . . . , X_n^±1] withR(T_n×C^×).

LetK^Gn×C×(Zn−1,n) be the convolution algebra with respect to the em- beddingZ_n−1,n ⊆Y_n×Y_n. Let

• Hn−1,n be the parabolic subalgebraHn−1⊗CC[X_n^±1] of Hn, and

• ι_n:Zn−1,n ⊆Z_n be the inclusion map.

We attribute the next theorem to Ginzburg [6] and Lusztig [22]. In [16], it was stated as an isomorphism of bimodules.

Theorem 3.4.

(1) We have an isomorphism ofR(G_n×C^×)-algebras H_n≃K^Gn×C×(Z_n)by the above choice of T_i andX_i in K^Gn×C×(Z_n).

(2) The inclusion mapι_ninduces the following commutative diagram ofZ(H_n)- algebras.

ι_n∗ :K^Gn×C×(Z_n−1,n) → K^Gn×C×(Z_n)

↓ ↓

Hn−1,n ⊆ H_n where the vertical arrows are isomorphisms.

It is also clear that the inclusion mapY_n≃Z_n(e)֒→Z_n−1,n induces K^Gn×C×(Y_n)→K^Gn×C×(Z_n−1,n)

and it is identified withR(T_n×C^×)֒→H_n−1,n. 3.2. The embedding of Hn−1 into H_n. Let

Y_n−1,n =T^∗Fℓ_n|Fℓn−1,

where we identify Fℓn−1={F ∈ Fℓ_n|Fn−1 =Cⁿ⁻¹}, and let Nn−1,n ={X ∈ Nn|XCⁿ⁻¹⊆Cⁿ⁻¹}.

Then we define

Z_n−1,n^′ =Yn−1,n×N_n−1,nYn−1,n

={(X, F, F^′)∈Z_n|F_n−1 =F_n−1^′ =Cⁿ⁻¹}.

LetPn−1,nbe the maximal parabolic subgroup ofG_nthat stabilizesCⁿ⁻¹. The Levi partL_n−1,n×C^×ofP_n−1,n×C^×is (G_n−1×C^×)×C^×, which acts on Zn−1by letting the middle component act trivially. We denote the unipotent radical ofPn−1,n byUn−1,n. It is also the unipotent radical ofPn−1,n×C^×. Explicitly,

Ln−1,n=

Gn−1 0

0 C^×

, Un−1,n=

1n−1 ∗

0 1

. We consider the following diagram.

Z_n−1,n^{′ µ}←−^n−1,n (Gn×C^×)×Z_n−1,n^{′ νn−1,n}−→ (Gn×C^×)×_Pn−1,n×C^×Z_n−1,n^′ =Zn−1,n. Then we have the restriction map

Res^G_P^n×C×

n−1,n×C^×:K^Gn×C×(Zn−1,n)≃K^{Pn−1,n×C×}(Z_n−1,n^′ ).

Z_n−1,n^′ is aL_n−1,n×C^×-equivariant vector bundle of rankn−1 overZ_n−1and we writeκn−1,n :Z_n−1,n^′ → Zn−1. Thenκ^∗_n−1,ngives the Thom isomorphism

K^{Ln−1,n×C×}(Z_n−1,n^′ )←^∼ K^{Ln−1,n×C×}(Zn−1).

Noting that

K^{Pn−1,n×C×}(Z_n−1,n^′ )≃K^{Ln−1,n×C×}(Z_n−1,n^′ )

by the forgetful map, and

K^{Ln−1,n×C×}(Zn−1)≃K^Gn−1×C×(Zn−1)⊗CC[X_n^±1], we have

K^{Pn−1,n×C×}(Z_n−1,n^′ )≃K^Gn−1×C×(Zn−1)⊗CC[X_n^±1].

Now, the following holds.

Proposition 3.5. We have the following isomorphism of R(L_n−1,n×C^×)- algebras

K^Gn×C×(Zn)⊇K^Gn×C×(Zn−1,n)≃K^Gn−1×C×(Zn−1)⊗CC[X_n^±1], which gets identified with H_n⊇H_n−1,n=H_n−1⊗^CC[X_n^±1].

Proof. We only have to show that b_i 7→b_i, for 1≤i < n−1, andθ_λ 7→ θ_λ. Define

Z_n−1,n^′ (s_i) ={(X, F, F^′)∈Z_n−1,n^′ |F_j =F_j^′ for all j6=i, XF_i+1⊆F_i−1}.

Then ν_n−1,n⁻¹ (Z_n(s_i)) = (G_n×C^×)×Z_n−1,n^′ (s_i) and we have ν_n−1,n^∗ O_Zn(s

i)=µ^∗_n−1,nO_Z′

n−1,n(si), O_Z′

n−1,n(si) =κ^∗_n−1,nO_Zn−1(s

i). Hence,b_i 7→b_i, for 1≤i < n−1.

LetZ_n−1,n^′ (e) ={(X, F, F^′)∈Z_n−1,n^′ |F =F^′}and consider the diagram ν_n−1,n⁻¹ (Z_n(e)) = (G_n×C^×)×Z_n−1,n^′ (e)

ւ ↓νn−1,n

Z_n−1,n^′ (e)≃Y_n−1,n ⊆ Y_n≃Z_n(e)

πn−1,n↓ ↓πn

Fℓn−1 ⊆ Fℓ_n Then ν_n−1,n^∗ π_n^∗L_λ=µ^∗_n−1,nπ^∗_n−1,nL_λ|Fℓ_n−1 and

L_λ|Fℓn−1 = (P_n−1,n×C^×)×_Bn×C^×C_λ. But the diagram

Z_n−1,n^′ (e) =κ⁻¹_n−1,n(Z_n−1(e))≃Y_n−1,n ^κ−→^n−1,n Y_n−1 ≃Z_n−1(e) ց ↓π_n−1

Fℓn−1

shows

π^∗_n−1,nL_λ|Fℓn−1 =κ^∗_n−1,n π_n−1^∗ ((P_n−1,n×C^×)×_Bn×C^×C_λ) . Hence,θ_λ 7→θ_λ, forλ∈Hom(T_n,C^×).

As the generatorsb_i andθ_λ correspond correctly, it is an isomorphism of R(Ln−1,n×C^×)-algebras, which is identified withHn−1⊗C[X_n^±1]֒→H_n.

3.3. Specialized Hecke algebras. Let ζ ∈ C be a primitive e^th root of unity, for e ≥ 2. We fix a diagonal matrix s = diag(ζ^s1, . . . , ζ^sn), and set a= (s, ζ)∈G_n×C^×. We denote byAthe smallest closed algebraic subgroup of G_n×C^× that contains a, namely the cyclic group hai of order e in our case. Note that A is contained in (Gn−1×C^×)×C^×.

Definition 3.6. We denote the A-fixed points of M by M^a, for M = Z_n, Zn−1,n,Yn=T^∗Fℓn, Fℓn etc.

LetC_abe theR(T_n×C^×)-module defined byX_i 7→ζ^si, for 1≤i≤n, and q 7→ζ. C_a|_R(Gn×C^×) defines a central character Z(H_n) =R(G_n×C^×)→C. Then we writeC_a⊗_Z(Hn)−, for the specialization of the center with respect to the central character. We define f_a∈C[X_n] by

f_a(X_n) = (X_n−ζ^s1)· · ·(X_n−ζ^sn).

Definition 3.7. TheC-algebraH_n^a=C_a⊗_Z(Hn)H_nis called thespecialized Hecke algebra of rank n at a. The specialized algebra C_a⊗_Z(Hn)Hn−1,n of the parabolic subalgebra Hn−1,n is denotedH_n−1,n^a .

Lemma 3.8. Leta_kbe thek-th elementary symmetric function inX₁, . . . , X_n evaluated at X₁ = ζ^s1, . . . , X_n = ζ^sn. Then, C[X_n^±1]/(f_a) is a Z(H_n−1)- algebra via

e_k7→a_k−a_k−1X_n+· · ·+ (−1)^kX_n^k,

where e_k is the k-th elementary symmetric function in X₁, . . . , X_n−1, and H_n−1,n^a =H_n−1⊗_Z(Hn−1)C[X_n^±1]/(f_a).

Proof. Ase_k+X_ne_k−1is thek-th elementary symmetric function inX₁, . . . , X_n, the surjective map

H_n−1,n =H_n−1⊗^CC[X_n^±1]→H_n−1⊗_Z(Hn−1)C[X_n^±1]/(f_a)

factors throughH_n−1,n^a . On the other hand, both have the same dimension

n!(n−1)!. Hence the result.

As Hn ≃ K^Gn×C×(Zn) and Hn−1,n ≃ K^Gn×C×(Zn−1,n) as R(Gn×C^×)- algebras, we identify the following C-algebras respectively.

H_n^a=C_a⊗_R(Gn×C^×)K^Gn×C×(Z_n), H_n−1,n^a =C_a⊗_R(Gn×C^×)K^Gn×C×(Zn−1,n).

By Proposition 3.5, geometric realization of Lemma 3.8 is given by C_a⊗_Z(Hn)K^Gn×C×(Z_n−1,n)≃C_a⊗_Z(Hn)

K^Gn−1×C×(Z_n−1)⊗C[X_n^±1]

≃K^Gn−1×C×(Zn−1)⊗_Z(Hn−1)C[X_n^±1]/(f_a).

Letm_i be the multiplicity of ζⁱ in{ζ^s1, . . . , ζ^sn}. Then C[X_n^±1]/(f_a)≃ M

i∈Z/eZ

C[X_n^±1]/((X_n−ζⁱ)^mi).

Definition 3.9. We denote by p_i the identity of C[X_n^±1]/((X_n −ζⁱ)^mi) which is viewed as an element of H_n−1,n^a . Thus, p_i are central idempotents of H_n−1,n^a such thatP

i∈Z/eZp_i = 1 andp_ip_j =p_jp_i =δ_ijp_i. We have the decomposition ofC-algebras

H_n−1,n^a = M

i∈Z/eZ

p_iH_n−1,n^a p_i.

We fixi∈Z/eZand suppose that (ν₁, . . . , ν_n) is a permutation of (s₁, . . . , s_n) such thatνn=i. Then

(diag(ζ^ν1, . . . , ζ^νn−1), ζ)∈G_n−1×C^×

defines a central character ofHn−1 and we may define the specialized Hecke algebra with respect to the central character. We denote it by H_n−1^a;i . By Lemma 3.8, we have the surjective algebra homomorphism

p_iH_n−1,n^a p_i−→H_n−1^a;i ,

because if we write b_k=a_k−a_k−1ζⁱ+· · ·+ (−1)^kζ^ki, then

n−1X

k=0

(−1)^kb_kT^k

!

(1−ζ^νnT) = Xn k=0

(−1)^k(b_k−1ζⁱ+b_k)T^k

= Xn k=0

(−1)^ka_kT^k

= (1−ζ^s1T)· · ·(1−ζ^snT)

ande_k7→b_k, for 1≤k≤n−1, is the central character which definesH_n−1^a;i . Composing it with the projection H_n−1,n^a to p_iH_n−1,n^a p_i, we have

H_n−1,n^a −→H_n−1^a;i ,

which is nothing but the specialization map at X_n = ζⁱ. Its geometric realization is given by

C_a⊗_Z(Hn)K^Gn×C×(Z_n−1,n)−→C_a;i⊗_Z(Hn−1)K^Gn−1×C×(Z_n−1).

Lemma 3.10. Simple p_iH_n−1,n^a p_i-modules are obtained from simple H_n−1^a;i - modules through the algebra homomorphism p_iH_n−1,n^a p_i→H_n−1^a;i .

Proof. Let I_a be the two-sided ideal of p_iH_n−1,n^a p_i generated by X_n −ζⁱ. Then I_a is nilpotent, so that I_a acts as zero on simple p_iH_n−1,n^a p_i-modules.

AsH_n−1^a;i =piH_n−1,n^a pi/Ia by Lemma 3.8, we have the result.

Ginzburg’s theory tells us how to realize the specialized Hecke algebra in sheaf theory. Definitions of the maps in (1) are necessary in the proof of (2), so that they will be given in the proof.

Theorem 3.11.