korteweg-de vries instituut universiteit van amsterdam · introduction harmonic analysis...

Introduction Harmonic analysis Noncommutative geometry

Hecke algebras and harmonic analysis

Eric Opdam

Korteweg-de Vries InstituutUniversiteit van Amsterdam

August 2006, ICM Madrid


Outline

1 IntroductionOrigin and Motivation

2 Harmonic analysisPlancherel formula for p-adic reductive groupsThe Plancherel formula for affine Hecke algebras

3 Noncommutative geometryK-theory and index functions


Origin and Motivation


Hecke algebras were introduced by Shimura (1950’s) inmodern abstract setting, modelled after the famous work ofHecke (1930’s) on Euler products for automorphic forms.



Applicability of Hecke algebras

Hecke algebras have turned out to be widely applicable.Besides in number theory Hecke algebras arise in variousareas of representation theory, in combinatorics, in the theoryof knots, quantum groups, integrable models, orthogonalpolynomials. This represents the work of many mathematicianswith a variety of backgrounds (see the references in theproceedings article...).



Harmonic analysis and Hecke algebras

In this talk I will discuss the natural application of Heckealgebras in the harmonic analysis of p-adic reductivegroups.This places affine Hecke algebras in an analytic context.As we will see, this gives rise to new problems, but also itsheds new light on old solutions.What we are to discuss here is relevant to various otherapplications of Hecke algebras as mentioned above, oreven in a more general context.


Plancherel formula for p-adic reductive groups


The main problem of harmonic analysis on a locally compact,unimodular, type I group G (e.g. a p-adic reductive group):

Plancherel Formula

δ =

∫π∈G

χπdµ(π)

δ is the delta distribution at e ∈ G.µ is called the Plancherel measure of G. It is a positive Radonmeasure on G, unique up to normalization of the Haar measureon G.



Plancherel formula for p-adic groups

The Plancherel formula presents a hierarchy of verychallenging partial problems.

Describe the unitary dual G of G, or at least the support ofthe Plancherel measure Supp(µ) ⊂ G.Determine the irreducible characters of G.Compute the Plancherel measure µ.



Splitting the Plancherel formula in blocks

Hecke algebras assist by breaking up this problem in moremanageable blocks.

For finite reductive groups this was exploited verysuccessfully (Howlett, Lehrer, Lusztig,...).In this talk we want to study this method to p-adic groups.For a reductive p-adic group G the main underlying idea forthis to work is the Bernstein decomposition theorem of thecategory of smooth representations of G.



Describing blocks by affine Hecke algebras

Due to the work of many (e.g. Borel, Bushnell, Kutzko, Morris,Lusztig, Prasad, Moy,...) the Bernstein blocks are often knownto be equivalent to the representation category of certainexplicit affine Hecke algebras, the so-called affineIwahori-Matsumoto Hecke algebras or simply affine Heckealgebras.


The Plancherel formula for affine Hecke algebras

Affine Weyl groups

In this talk W will be an extended affine Weyl group. W actsdiscretely by isometries on a Euclidean space V , and

W acts transitively on the set of alcoves (closed chambers)of an affine root hyperplane arrangement in V .We choose a fundamental alcove C. The number ofseparating affine hyperplanes between C and w(C) iscalled the length l(w) of w ∈ W .W contains the set S of reflections in the walls of C.S ⊂ W is called the set of affine simple reflections.We put W a = 〈S〉 C W , and Ω ⊂ W for the subgroup oflength 0 elements. Then W = W a o Ω.



W contains a lattice of translations X ⊂ V as a normalsubgroup.The isotropy subgroup W0 of 0 ∈ V is the Weyl group of afinite integral root system R0 ⊂ X .W is a semidirect product W = W0 n X .



Definition of the affine Hecke algebra

The affine Hecke algebra H is the complex, unital associativealgebra with basis Nw (w ∈ W ) and relations:

If u, v ∈ W and l(uv) = l(u) + l(v) then NuNv = Nuv .For all affine simple reflections s ∈ S:

(Ns − q(s)1/2)(Ns + q(s)−1/2) = 0

for certain positive real numbers q(s) (s ∈ S) such thatq(s) = q(s′) if s ∼ s′ in W .

The affine Hecke algebra H(W ,q) is a deformation of thegroup algebra C[W ] of the affine Weyl group, in thepositive parameters q(s).



Bernstein-Zelevinski presentation of H

Theorem (Bernstein, Zelevinski, Lusztig)Let T be the complex algebraic torus with character lattice X .There exists an injective algebra homomorphism θ : C[T ] → Hsuch that H is generated by the commutative subalgebraA = θ(C[T ]) ⊂ H and by the elements Ns with s ∈ S0 (thesimple reflections in W0). The cross relations:

Ns.a− as.Ns = (q(s)1/2 − q(s)−1/2)a− as

1− θ−α

Corollary: the center of HThe center Z of H is equal to Z = AW0 = C[T ]W0 .



H is a ∗ algebra with distinguished tracial state.

The anti-linear anti-involution ∗We define an anti-linear anti-involution on H by N∗

w := Nw−1 .

The distinguished trace τ on HWe define τ(Nw ) = δw ,e.

Positivity: τ is positive, hence a tracial stateThe basis Nw is orthonormal with respect to the Hermitian form(x , y) = τ(x∗y) on H. In particular, τ is a positive tracial stateon the ∗-algebra (H, ∗).



The Plancherel formula for H(W , q)

Abstract Plancherel formula (Matsumoto)

There exists a compact topological space H of irreducible∗-representations of H, and a unique positive measure µPl onH such that

τ =

∫π∈H

χπdµPl(π)

There is a finite-to-one map z : H → W0\T .



Eisenstein functionals

Weyl denominator and q-Weyl denominatorWe define

D(t) =∏

α∈R0,+

(1− α(t)−1)

Dq(t) =∏

α∈R0,+

(1− q−1α∨α(t)−1)

Eisenstein functionals on HThere exists a unique holomorphic function E : T → H∗ (H∗ isthe linear dual of H) such that Et(1) = D(t) and such that for allt ∈ T :

Et(θxh) = Et(hθx) = x(t)Et(h)



Integral formula for the trace τ of H

Contour integral for the trace

Write T = Tv Tu for the polar decomposition of T in a vectorgroup Tv ' Rn and a compact torus Tu ' (S1)n. Then

τ(h) = q(w0)−1

∫t∈pTu

Et(h)

D(t)D(t)D(t−1)

Dq(t)Dq(t−1)dt

where p ∈ Tv is deep in the negative chamber of Tv .



The poles of the kernel

We would like to refine the contour integral to a spectraldecomposition of τ , or at least the diagonalization of the centerin the Hilbert space completion L2(H) of H. This is acomputation of residue distributions.

Residual cosetsA residual coset L ⊂ T is a coset of a (complex) subtorus of Talong which the pole order oL of

D(t)D(t−1)

Dq(t)Dq(t−1)

is at least equal to codim(L).



Existence and Uniqueness of Residues

Tempered real form of a residual coset

The tempered part Lt of a residual coset L is the uniquecompact form of L such that its projection cL to Tv has minimaldistance to e ∈ Tv . We call cL the center of L.

Theorem (with Heckman)

Let h ∈ H. There exist unique distributions XhL on cLTu

supported on Lt ⊂ cLTu such that for all a ∈ A = C[T ]

τ(ha) =∑

L residual

XhL(a|cLTu)



From distributions to positive measures

Positivity Theorem

The restriction of X1 =∑

L X1L to Z = AW0 = C[T ]W0 (the center

of H) defines a positive, W0-invariant measure ν = X1|Z on T ,supported in the union S = ∪LLt of the tempered residualcosets.

Continuity Theorem

For any h ∈ H we have: Xh =∑

L XhL is defines a complex

measure which is absolutely continuous with respect to ν.

Support Theorem (no cancellation of residues)

Supp(ν) = S = ∪LLt



Implications for Root Systems

Bound on Pole Order (to appear)For any coset L of a subtorus of T the pole order oL along L isat most equal to codim(L). Explicitly:

oL := |α ∈ R0 | α|L = qα∨| − |α ∈ R0 | α|L = 1| ≤ codim(L)

Parabolic induction of residual cosetsLet L be a residual coset, and let RL ⊂ R0 be the subset ofroots which are constant on L. Then rank(RL) = codim(L) andthere exists a RL-residual point (i.e. a zero dimensional residualcoset) rL ∈ TL such that L = rLT L.



Implications for Root Systems

W0-orbits of residual points are hermitian (to appear)

Let r ∈ T be a residual point. Then r−1 ∈ W0r .

No embedded tempered residual cosets (to appear)

If L,M are residual cosets and L 6= M then M t 6⊂ Lt

(equivalently, the spectral measure νL on Lt is smooth).



Diagonalization of the center Z in L2(H)

Spectral decomposition for H as Z-moduleWe have

τ(h) =

∫t∈S⊂T

χt(h)dν(t)

where:t → χt(h) is a W0-invariant function on S.For all t ∈ S, χt is a positive tracial state with centralcharacter W0t .χt is a positive linear combination of (finitely many)irreducible tempered characters of H with central characterW0t .On each Lt , ν is a smooth measure, given by an explicitproduct formula.



Unknowns: the set of discrete series at residual points

Discrete series charactersFor each W0-orbit W0r ∈ T of residual points let ∆W0rdenote the (finite, nonempty) set of equivalence classes ofirreducible discrete series representations (δ,Vδ) (withχδ ∈ L2(H)), and dδ ∈ R+ such that

χr =∑

δ∈∆W0r

dδχδ

If we put q(s) = qfs for fixed fs ∈ R (for s ∈ S) then thenumbers dδ > 0 are independent of q > 1 (we say: dδ isindependent of scaling isomorphisms).



Residual points and distinguished unipotent orbits

Equal parameter caseIn the equal parameter case with X = P (the weight lattice)there is a natural bijection between the set of W0-orbits ofresidual points and LG-conjugacy classes of pairs(s,u) ∈ LG × LG with s ∈ LG semisimple such that thecentralizer CLG(s) is semisimple, and u ∈ CLG(s) distinguishedunipotent.

Theorem (Kazhdan-Lusztig)

Let W0r correspond to the orbit of (s,u).Then ∆W0r is in bijection with the set of geometric irreduciblecharacters of CLG(s,u)/ZCLG(s,u)0.



Formal dimension product formula

Product formula for formal dimensionsThe Plancherel measure of δ ∈ ∆W0r equals:

µPl(δ) = dδν(W0r) = dδλW0r q(w0)

∏′α∈R0

(α(r)− 1)∏′α∈R0

(qα∨α(r)− 1)

where λW0r ∈ Q and where∏′ denotes the product in which we

omit the factors which are equal to 0.



Application of product formula: formation of L-packets

The product formula of ν combined with the scale invariance ofthe dδ were used to determine the unipotent discrete seriesL-packets for the exceptional p-adic Chevalley groups of adjointtype (Heckman-O. (based on conjectures which have beensolved now), Reeder). Lusztig has determined the L-packets ingeneral for p-adic Chevalley groups of adjoint type by othermethods.



Groupoid of induction data

H determines a compact orbifold groupoid WΞ whose setof objects Ξ consists of parabolic induction dataξ = (RP , δ, t) with RP ⊂ R0 a standard parabolicsubsystem, with δ ∈ ∆(HP) an irreducible discrete seriescharacter for HP = H(XP ,RP ,YP ,R∨P ), and with t ∈ T P

u .The set of arrows Wξ,η from ξ = (P, δ, t) ∈ Ξ toη = (Q, δ′, t ′) ∈ Ξ in the groupoid WΞ are given by twistingof ξ with Weyl group elements w (and by elements acertain finite abelian group) such that w(ξ) = η (inparticular, w(P) = Q).There is a natural finite surjective continuous map

|Ξ| := W\Ξ → W0\S ⊂ W0\T(P, δ, t) → W0(rδt)



The Fourier transform

Plancherel formula and Fourier transformThere exists a smooth projective unitary representation π of thegroupoid WΞ in Modfd

unitary(H), and a unique measure µPl suchthat the Fourier transform

F : L2(H) → L2(Ξ,End(π), µPl)W

h → ξ → π(ξ)(h)

is an isometric isomorphism of H×H-bimodules.



Fourier isomorphism on The Schwartz algebra

Theorem/Definition Schwartz algebra

We define S = s =∑

w∈W cwNw | ∀n ∈ N : pn(s) :=supw∈W|cw |l(w)n <∞. This a Fréchet algebra with respectto the seminorms pn.

Theorem (Delorme-O.): Support of Plancherel measure

The support of µPl is the set S of irreducible temperedrepresentations of H, i.e. those irreps which extendcontinuously to S.



The Schwartz algebra

Theorem (Delorme-O.)The Fourier isomorphism restricts to an isomorphism

FS : S → C∞(Ξ,End(π))W (1)

Corollary: The center ZS ⊂ SThe center ZS = C∞(Ξ)W of the algebra S is the algebra ofsmooth functions on the orbifold WΞ.

CorollaryFor every irreducible tempered representation ρ there is aunique orbit Wξ so that ρ is equivalent to a summand of π(ξ).We call Wξ the tempered central character zS(ρ) of ρ.



Support of Plancherel measure

Fundamental problem: The support of µPl

Determine (parametrize) Supp(µPl) ⊂ H.



Reduction to discrete series

Let ∆ = ∪P(P,∆(HP)) (P standard parabolic subset), with∆(HP) the set of irreducible discrete series of HP . Then W actson this set by twisting, and this defines a finite groupoid W∆.

If we would be interested in Supp(µPl) only up to µPl 0-setsthen it is enough to determine W∆, because thisdetermines WΞ.Problem A’: Determine the finite groupoid W∆.To describe Supp(µPl) precisely we will need slightly moreinformation: Choose a realization (Vδ, δ) for each class(P, [δ]) in ∆, and an intertwining isomorphismδw : Vδ → Vδ′ for each twisting isomorphism w : HP → HQ.This yields a projective unitary representation δ of W∆,unique up to natural isomorphisms. Let η∆ ∈ H2(W∆,C×)be its cohomology class.Problem A: Determine W∆ and η∆.




Explicit answers to A are known in special cases, inparticular in the case of equal parameters and X = P, bythe work of Kazhdan and Lusztig.Lusztig has solved A (in principal at least) for more generalcases, by geometric methods.




Definition: Analytic R-groups (Delorme-O.)

For each orbit ξ ∈ Ξ we define (up to isomorphism) an analyticR-group Rξ ⊂ Wξ,ξ depending on the density of µPl at ξ.In addition we define a 2-cocycle ηξ for Rξ in terms of η∆ (recallProblem A).

Knapp-Stein type Theorem (Delorme-O.)

The module π(ξ) is a unitary S − C[Rξ, ηξ]-bimodule whichdefines a Morita equivalenceEξ : Mod(C[Rξ, ηξ]) → ModWξ,unitary(S).

Reduction to discrete seriesThe above Theorem reduces the problem to describeSupp(µPl) completely to Problem A.



Fundamental problem: Computation of µPl

The problem to compute µPl reduces to

Computation constants dδ(HP)

Problem B: For (P, δ) ∈ ∆, compute the positive real constantsdδ(HP).

Answers to B are known for exceptional root systems in theequal par. case (M. Reeder). In the general equal parametercase there is a conjecture for their value; in general not eventhat.

ConjectureThe dδ are rational numbers.



Outline

The remaining part of this talk

In the remaining part of this talk I want to discuss conjectures innoncommutative geometry which might help to solve problemsA and B. This approach is different from the algebro-geometricapproach but based on deformations in the parameters q(s).We expect this to be especially useful (effective) in unequallabel cases.We will assume (for simplicity) from here on that the root datumis semisimple, i.e. that QR0 = Q⊗ X .


K-theory and index functions

The Euler-Poincaré map

Theorem (O., Reeder) (unpublished)Every H-module of finite length has homological dimension≤ dimC(T ). In fact we have defined an explicit functorF : Modfl(H) → Kb(H) (the category of bounded complexes ofprojective H-modules), together with a projective resolutionF•(V ) → V → 0.

Change of base ring

Let K0(H) be Grothendieck group of the finitely generatedprojective H modules, and let G(S) be the Grothendieck groupof the S-modules of finite length. We have naturalhomomorphisms β : K0(H) → K0(S) (change of base ring) andρ : G(S) → G(H) (forget tempered).



Tempered elliptic representations

We denote by

Euler-Poincaré mapWe define an anti-linear Euler-Poincaré mapε : GC(S) → K C

0 (S) such that ε(V ) = β(∑

i(−1)i [Fi(ρ(V ))]) forall V ∈ Modfl(S).

Definition

Let GCI (S) denote the subspace of GC(S) generated by the

modules induced form proper parabolic subquotient algebrasHP . We define Elltemp := GC(S)/GC

I (S), a ZS-module.



Tempered elliptic representations and the R-group

Theorem (after Arthur)The equivalence Eξ induces a linear isomorphismG(Eξ) : GC(Rξ, ηξ) → GC(S)Wξ. Via G(Eξ) the specializationElltempWξ = Elltemp⊗ZSCWξ corresponds to the space Ell(Rξ, ηξ)

of η-twisted class functions of Rξ which have support on the setof elliptic conjugacy classes of Rξ, i.e. the elements r ∈ Rξ forwhich ξ is an isolated fixed point.

Corollary

The space Elltemp is a finite dimensional semisimpleZS = C∞(Ξ)W -module.



The index pairing

Definition of the index pairing

There is a natural “index pairing” [·, ·] : K0(S)×G(S) → Zdefined as follows: given an idempotent p ∈ Mn(C)⊗ S and a(V , π) ∈ Modfl(S) we set [p, [π]] := rank((Id⊗π)(p)).

The discrete part K discr0 (S) ⊂ K C

0 (S) of K0(S)

We define the support of a class [α] ∈ K C0 (S) to be the subset

Wξ ∈ W\Ξ | ∃π ∈ GCWξ(S) : [α, [π]] 6= 0.

We define K discr0 (S) ⊂ K C

0 (S) to be the subspace of classeswhose support has dimension 0.



The index pairing

Theorem

The anti-linear map ε factors through ε : Elltemp → K discr0 (S).

Hermitian pairing on Elltemp

We define a Hermitian pairing 〈·, ·〉el on Elltemp by〈χ, ψ〉 := [ε(χ), ψ].



The index pairing

Conjecture A (“Arthur type formula”)

Let φ, ψ ∈ Ell(Rξ, ηξ) be elliptic twisted characters of Rξ. Then

〈G(Eξ)(φ),G(Eξ)(ψ)〉el = |Rξ|−1∑r∈Rξ

det(1− r)φ(r)ψ(r)

Elliptic representations with distinct tempered central characterare orthogonal.

RemarkThe pairing on Ell(Rξ, ηξ) is positive definite (since the supportof the function det(1− r) is exactly equal to the set of ellipticconjugacy classes of Rξ).



This conjecture is the analogue of well known results forp-adic groups.Kazhdan conjectured that the elliptic pairing of ellipticcharacters of a p-adic reductive group G can be rewrittenas integral over the set of regular semisimple ellipticconjugacy classes of G with respect to a canonicalmeasure dµ.The Kazdan conjecture was proved by Bezrukavnikov, andindependently by Schneider-Stuhler.Arthur derived the pairing in terms of analytic R-groupsfrom this integral over the elliptic classes, using the localtrace formula.For the Hecke algebra H the intermediate step, the integralover the regular elliptic conjugacy classes make no sense.Also these results are based on the theory of orbitalintegrals, which is not available for Hecke algebras.



Corollary

The Hermitian pairing 〈·, ·〉el on Elltemp is positive definite.

Corollary

ε : Elltemp → K discr0 (S) is an anti-linear isomorphism.

Corollary

The irreducible discrete series representations (U, δ) ∈ ∆(H)give rise to an orthonormal set of vectors [U] in Elltemp.

Index function

Let (U, δ) ∈ Modfl(S). There exists an index function fU ∈ H forU such that for any virtual tempered character χ ∈ GC(S) wehave 〈χU , χ〉el = χ(fU).



L2-index (Euler-Poincaré identity)

If (U, δ) ∈ ∆W0r then

µPl(δ) = dδν(W0r)

= dδλW0r q(w0)

∏′α∈R0

(α(r)− 1)∏′α∈R0

(qα∨α(r)− 1)

= τ(fU)

=∑

f

(−1)dim(f )∑

σ∈Irr(Hf oΩf )

[U|(Hf oΩf ) ⊗ εf : σ]dσ(q)

where f runs over a complete set of representatives of theΩ-orbits of faces of the alcove C, and where dσ(q) ∈ Q(q1/2

s )denotes the formal dimension of σ in the finite dimensionalHilbert algebra Hf o Ωf whose trace is the restriction of thetrace τ of H.



Conjecture A for 〈·, ·〉el applies to problem B

By Euler-Poincaré identity, we see that dδ ∈ Q+.



Towards problem A: Finding the support of µPl

Assume that q(s) = qfs for all s ∈ S and fixed fs ∈ R. Givena tempered irreducible representation (V , π) of H, thereexists a unique continuous family (V , π(q)) (with q > 1) oftempered irreducible representations of H(q) such that theunitary part of the central character of π(q) is independentof q. The limit π(1) exists and is a tempered representation(not semisimple, in general) of H(1) = C[W ].This defines a linear map evq→1 : G(S(q)) → G(S(W )),preserving all “K-types” (Tits’ deformation lemma) andsending GI(S(q)) to GI(S(W )).

Consequence of Conjecture A

The induced map evq→1,el is isometry for 〈·, ·〉el , and thusinjective on Elltemp(q).



Towards problem A: Describing the support of µPl

Maarten Solleveld has defined a homomorphismα(q) : K0(S(W )) → K0(S(q)), using the continuous family ofFréchet algebras S(q) and the fact that the Fréchet algebrasS(q) and S(q′) are isomorphic for any q,q′ > 1.

Consequence of Conjecture A

We have αC(q)(K discr0 (SW )) ⊂ K discr

0 (S(q)), andε−1q αC(q) εq=1 is the adjoint of evq→1,el . In particular, this

restriction of αC(q) to K discr0 (SW ) is onto K discr

0 (S(q))



Conjecture B: The invariance of the groups K0(S(q))

The linear map αC(q) is an isomorphism.

For equal parameters this follows from general results ofMaarten Solleveld combined with a Theorem of Baum andNistor for the periodic cyclic homology HP∗(H).

Solleveld proved that the Chern character yields anisomorphism ch : K C

∗ (S)∼−→ HP∗(S).

Solleveld also proved HP∗(S) ' HP∗(H).Baum and Nistor proved that HP∗(H) ' HP∗(C[W ]) (Hwith equal parameters). With Solleveld’s results thisimplies the conjecture for this case.For Hecke algebras of reductive p-adic groups thisconjecture is stated by Baum, Connes and Higson (1994)in relation to the Baum-Connes conjecture.



Conjecture A and Conjecture B imply

Conjecture C

evq→1,el : Elltemp(S(q)) → Elltemp(S(W )) (2)

is an isomorphism.

This was proved by Reeder for equal parameters and X = P. Itwas shown in greater generality by Waldspurger (I think).

Refinement of Conjecture C

The family q → Elltemp(S(q)) of semisimpleZ ' C[T ]W0-modules is continuous.



Same events for Elltemp



Towards problem A: Finding the support of µPl

Reduction to generic case

Assuming conjecture C (refined) we can reduce theclassification problem for Supp(µPl) in principle to the genericcase. This is a considerably simpler problem for the non simplylaced cases.



Example: The 3-parameter case Caffn

We assume X = Zn, R0 = ±ei ,±ei ± ej) The orbits ofresidual points for generic parameters q = (q1,q2,q3) areparametrized by ordered pairs (λ, µ) of partitions with|λ|+ |µ| = n (easy).On the other hand, the elliptic conjugacy classes of theR-groups for q = (1,1,1) (the group algebra of the affineWeyl group) are also parameterized by pairs (κ, ν) of|κ|+ |ν| = n (easy).But we know that each residual orbit carries at least oneirreducible discrete series representation!Hence Conjecture C implies: in the generic case, eachorbit of residual points carries one irreducible discreteseries representation.




For non generic q0: we can still easily parameterize theorbits W0r of residual points. But what about ∆W0r ?By Conjecture C (refined): the set ∆W0r is in bijection withthe set of generic residual orbits which coincide with W0rwhen we specialize at q = q0.This determines the action of W on ∆(q = q0).One easily proves that always η∆ = 1.Using the theory of the analytic R-group we can nowreconstruct the tempered dual.




It was verified by Klaas Slooten that this gives rise to thefamiliar combinatorial description of the Kazhdan-Lusztigparameters in terms of Lusztig’s family symbols whenq1 = q2 = q3 = q (which is highly nongeneric of course).For other “special values” of q, Slooten gave a similarcombinatorial description based on the generalizedsymbols of Lusztig and Shoji (in the real central charactercase).

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