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[ = ummation n action n ] q HHY fthe of (1.1) ions F ric q n e e , , As mentioned at the end of the introduction of [HHY, Appendix C], we have another candidate of q-analogue for (1.1): the q-Schur-Weyl duality between the quantum group U√q(sl2) and the Iwahori- 2 n Hecke algebra Hq(Sn) acting on the same tensor space (C )⊗ as (1.1), discovered by Jimbo [J86]. This one looks more natural in the representation theoretic viewpoint. However, there are several missing items in this q-Schur-Weyl duality to make a q-analogue of the arguments of [HHY]. One of them is the theory of Gelfand pairs and spherical functions for Iwahori- Hecke algebras. As sketched above, we used in [HHY] the corresponding theory for finite groups, for which we refer [M95, §7.1]. Since Iwahori-Hecke algebras are deformation of group algebras of Coxeter groups, it is natural to expect such a theory exists, although we could not find it in literature. The purpose of this note is to give the start line of the theory of Gelfand pairs and zonal spherical functions for Iwahori-Hecke algebras, which will be demonstrated in §3. The main objects introduced are Gelfand pairs in Definition 3.1 and zonal spherical functions in Definition 3.2. The main Theorem 3.3 gives fundamental properties of our zonal spherical functions, which can be regarded as a Hecke-analogue of the properties for finite groups [M95, VII.1, (1.4)]. Below is the list of notations used in the main text. (1) We denote by N := Z 0 = {0, 1, 2,...} the set of non-negative integers. (2) For a finite set S, we≥ denote by |S| the number of elements in S. (3) For a group G, we denote by e ∈ G the unit element.

2. Recollection on Iwahori-Hecke algebra As a preliminary of the main §3, we recall several basic notions on the structure and representation theories of Hecke algebras. Our main reference is [GP00]. Hereafter, Iwahori-Hecke algebras will be just called Hecke algebras.

2.1. Generic Hecke algebra. Here we explain our notation for the Iwahori-Hecke algebra,borrowing terminology and symbols from [GP00]. We make one major change on symbols: we denote the generic Hecke algebra by H, whereas it is denoted by KH in [GP00]. Let W be a finite Coxeter group with generating set S. We denote by ℓ(w) the length of a minimal expression of w ∈ W with respect to S. Let A be a unital commutative ring, and take a parameter set {as | s ∈ S}⊂ A such that as = at if s,t ∈ S are conjugate in W . We denote by

H = HA(W, S, {as | s ∈ S}) the Iwahori-Hecke algebra associated to the Coxeter system (W, S) over A, and simply call it the Hecke algebra. It is a unital associative A-algebra with generators {Ts | s ∈ S} and relations

2 Ts = (as − 1)Ts + as (s ∈ S), mst ((TsTt) = 1 (s,t ∈ S, s 6= t, mst < ∞).

The first relation is called the quadratic relation. In the second relation we denoted by ms,t the order of the element st in W . Let us recall some basic properties of H. For the detail and proofs we refer [GP00, 4.4.3, 4.4.6].

(1) For each w ∈ W , take a reduced expression w = si1 ··· sil . Then T := T ··· T ∈ H w si1 sil

is independent of the chose of the reduced expression. In particular, we have Te = 1 for the unit e ∈ W . Also the quadratic relation is rewritten as the factorized form (Ts + 1)(Ts − u) = 0. (2) For s ∈ S and w ∈ W , we have

Tsw (ℓ(sw) >ℓ(w)) TsTw = . ((as − 1)Tsw + uTw (ℓ(sw) <ℓ(w))

(3) As an A-module, H is free and finitely generated, and has an A-basis {Tw | w ∈ W }. We always assume as are invertible in A. Then the basis element Tw is invertible in H for any w ∈ W by [GP00, 8.1.1]. We recall two types of modules over the Hecke algebra H. The first one is the index representation given by

ind: H −→ A, Tw 7−→ aw (w ∈ W ), (2.1)

2 and the second one is the sign representation given by ℓ(w) sgn: H −→ A, Tw 7−→ (−1) (w ∈ W ). (2.2)

The well-definedness of these modules are shown by the quadratic relation (Ts + 1)(Ts − u) = 0. Next we cite from [GP00, §8.1] the specialization argument of the Hecke algebra. Let {us | s ∈ S} be 1 a set of indeterminates over C such that us = ut for s,t ∈ S conjugate in W . We set A := Z[u± ]. The generic Hecke algebra HA is defined to be the Hecke algebra over A:

HA := HA(W, S, {us | s ∈ S}).

Now assume that there is a ring map θ : A → C with θ(us) = q ∈ C \{0}, which will be called a specialization map. The map θ induces a C-algebra, which is denoted by

Hq := HA ⊗A C. Then by [GP00, 8.1.7], there exists a finite Galois extension K ⊃ C(u) such that the K-algebra

H := HA ⊗A K is semisimple and isomorphic to the group algebra K[W ]. Abusing the terminology, we also call H the generic Hecke algebra. Moreover, if the parameter q ∈ C \{0} is taken such that Hq is semisimple, then the specialization map H → Hq induces a bijection

Irr(H) −−→∼ Irr(Hq) (2.3)

between the sets of isomorphism classes of finite-dimensional simple modules over H and Hq respectively. In particular, for a semisimple Hq, the set Irr(Hq) is independent of q. It is known that for a generic complex number q, the algebra Hq is semisimple. A particular case is q = 1, where we have H1 = C[W ], and thus isomorphism classes of finite-dimensional simple H-modules correspond bijectively to those of finite-dimensional irreducible representations of W . Also note that the index representation (2.1) and the sign representation (2.2) correspond respectively to the trivial and sign representations of the Coxeter group W under the specialization q = 1. The generic Hecke algebra H over K is split semisimple in the sense of [GP00, Chap. 7]. Let us explain what it means. Since H is a semisimple K-algebra, there is a direct sum decomposition H =

V H(V ), where V runs over the simple H-modules, and each H(V ) is a simple K-algebra. Thus H(V ) is isomorphic to a matrix algebra Mn (DV ), where nV is the multiplicity of V in the decomposition L V of H as a module over itself, and DV is a division algebra over K. We also have DV ≃ EndH(V ) and dimK V = nV dimK DV . Now every simple H-module V is split simple. i.e., dimK DV = 1. This is the defining condition of split semisimplicity. Let us close this subsection by the case W = Sn with S = {s1,...,sn 1} being the set of transpositions − si = (i,i + 1). All of the generators si are conjugate in W , and the associated Hecke algebra H has a unique parameter q. Similarly, the generic Hall algebra H has a unique parameter u. The Hecke algebra Hq(Sn) mentioned in §1 is nothing but HC(W, S, q). As for the simple modules, the set Irr(H) of finite-dimensional simple H-modules sits in a series of bijections

Irr(H) −−→∼ Irr(H1) −−→{∼ λ ⊢ n}, (2.4) where λ ⊢ n means that λ is a partition of n. We will denote by Vλ the simple H-module corresponding to λ ⊢ n under (2.4). Note that the index representation (2.1) is V(n) and the sign representation sgn (2.2) is V(1n). 2.2. Schur elements. We continue to use the symbols in the previous subsection. Thus H denotes the Hecke algebra H associated to a finite Coxeter system (W, S) defined over a commutative ring A. We recall trace functions and their relation to the center Z(H) of H, referring [GP00, §7.1] for the detail. An A-linear map f ∈ H∗ := HomA(H, A) is called a trace function on H if f(hh′) = f(h′h) for any h,h′ ∈ H. We denote the A-module of trace functions on H by TF(H) := {f : H → A | trace functions on H}. H is a symmetric algebra in the sense of [GP00, 7.1.1], i.e., it is equipped with a trace function τ : H → A such that the A-bilinear form

H ⊗A H −→ A, h ⊗ h′ 7−→ τ(hh′) (2.5) is non-degenerate. Such a map τ is explicitly given by

τ : H −→ A, τ(Te)=1, τ(Tw)=0 (w 6= e).

3 We sometimes denote it by τH for the distinction.

For w ∈ W with a reduced expression w = s1 ··· sl, si ∈ S, we set aw := as1 ··· asl ∈ A. By [GP00, 8.1.1], we have

′ ′ 1 τ(Tw∨Tw )= δw,w ,Tw∨ := aw− Tw−1 ∈ H (w, w′ ∈ W ). (2.6)

Then for f ∈ H∗, we set

f ∗ := f(Tw)Tw∨ ∈ H. (2.7) w W X∈ By [GP00, 7.1.7], we have

f ∈ TF(H) ⇐⇒ f ∗ ∈ Z(H). (2.8) An important class of trace functions is given by the character of an H-module. Let us recall the definition. Let V be an H-module which is finitely generated and free as an A-module, and ρV : H → EndK (V ) be the corresponding A-algebra homomorphism. The character of V is defined to be the A-linear map

χV : H −→ A, χV (h) := trV ρV (h), where trV denotes the usual matrix trace. By the property of matrix trace, the character is a trace function, i.e., χV ∈ TF(H). Next we introduce Schur elements, following [GP00, §7.2]. Hereafter we consider the Hecke algebra H defined over a field K, instead of a commutative ring A. For H-modules V, V ′ and ϕ ∈ HomK (V, V ′), we define I(ϕ) ∈ HomK (V, V ′) by

I(ϕ)(v) := ϕ(vTw)Tw∨ (v ∈ V ). w W X∈ By [GP00, 7.1.10], we have I(ϕ) ∈ HomH (V, V ′). Now assume that V is a split simple H-module as explained in the previous subsection. Then by [GP00, 7.2.1], there is a unique element cV ∈ K such that

I(ϕ)= cV trV (ϕ) idV

for any ϕ ∈ EndK (V ), depending only on the isomorphism class of V . The element cV is called the Schur element associated to V . Let us explain representation theoretic meaning of Schur elements. Let V, V ′ be finite-dimensional simple H-modules. Then by [GP00, 7.2.2], we have

′ ′ χV (Tw)χV (Tw∨)= δV,V cV dimK V, (2.9) w W X∈ where δV,V ′ = 1 if V ≃ V ′, and δV,V ′ = 0 otherwise. Following [GP00, 8.1.8], we define the Poincar´epoly- nomial PW of the Coxeter group W to be the Schur element cind associated to the index representation ind (2.1) of the generic Hecke algebra H. By(2.9) and (2.6), it is given by N PW := cind = ind(Tw) ind(Tw∨)= ind(Tw) ∈ [us | s ∈ S]. (2.10) w W w W X∈ X∈ Schur elements are also related to idempotents of the generic Hecke algebra H, as shown in [GP00, 7.2.7]. Since H is split semisimple as explained in the previous §2.1, it has a decomposition H = H(V ) V Irr(H) ∈M into simple K-algebras H(V ), where V runs over finite-dimensional simple H-modules, and H(V ) ≃

MnV (K) with nV := dimK V . We have the corresponding decomposition of the unit 1 of H:

1= eV , eV ∈ H(V ). V Irr(H) ∈X Then eV are central primitive idempotents and mutually orthogonal, i.e., eV eV ′ = 0 if V 6≃ V ′. We have eV H ≃ V as H-modules. Moreover, by [GP00, 7.2.6], the Schur element cV is non-zero for any V ∈ Irr(H), and using the character χV we have 1 eV = χV (Tw)Tw∨. (2.11) cV w W X∈ 4 In particular, using (2.6) and (2.10), we can compute the idempotent eind corresponding to the index representation ind (2.1) as 1 eind = Tw. (2.12) PW w W X∈ For a finite-dimensional simple H-module V , let χV∗ ∈ H be the element associated to the character χV by the correspondence (2.7). Since χV is a trace function, we have χV∗ ∈ Z(H) by the equivalence (2.8). Moreover, by [GP00, 7.2.8], the elements {χV∗ | V ∈ Irr(H)} form a K-basis of Z(H), and

χV∗ = cV eV . (2.13)

The idempotence and orthogonality of eV yields

′ χV∗ χV∗ ′ = δV,V cV χV∗ (V, V ′ ∈ Irr(H)). (2.14)

Let us consider the behavior of objects discussed so far under the specialization H → Hq=1 = C[W ] C 1 induced by K → , u 7→ 1. Obviously we have Tw 7→ w and Tw∨ 7→ w− for each w ∈ W . For a finite-dimensional H-module V , we denote by Vq=1 the representation of W induced by the bijection (2.3) of finite-dimensional simple modules. Then by [GP00, Example 7.2.5], the specialization yields |W | cV 7−→ . dimC Vq=1

Thus we may say that the Schur element cV is a “q-analogue” of the quantity |W | / dim Vq=1. The

relation (2.9) reduces to the orthogonal relation for characters χVq=1 of the finite group W : 1 ′ 1 ′ χVq=1 (w)χV (w− )= δV,V . |W | q=1 w W X∈ The Poincar´epolynomial PW = cind (2.10) reduces to |W |, and the idempotent eV (2.11) for a simple H-module V reduces to

dimC Vq=1 1 e =1 = χ =1 (w)w− , Vq |W | Vq w W X∈ which is the primitive idempotent associated to the irreducible representation Vq=1 of W . In particular, 1 the idempotent eind for the index representation reduces to |W |− w W w, the idempotent of the group algebra C[W ] corresponding to the trivial representation of W . ∈ P We close this subsection by an explicit formula of Schur elements in the case W = Sn. Recall (2.4) that finite-dimensional simple H-modules are parametrized by partitions of n. Let λ = (λ1, λ2,... ) be such a partition. We denote by ℓ = ℓ(λ) the length of λ, i.e., the number of non-zero λi’s. Let Vλ be the simple H-module corresponding to λ, and cλ ∈ K be the Schur element associated to Vλ. By [GP00, 9.4.5, 10.5.1, 10.5.2], we have

1 n(λ) cλ− = u λi − λj u λi u! 1 i

1 dim Vλ,q=1 c− = λ − λ λ != , λ u 1 i j i |W | → 1 i

hf,giH∗ := τ(f ∗g∗) (f,g ∈ H∗). (2.15) Then (2.9) is rewritten by the pairing (2.15) as

′ ′ hχV ,χV iH∗ = cV δV,V , 5 3. Gelfand pairs and zoal spherical functions for Iwahori-Hecke algebras We give a Hecke analogue of the theory of Gelfand pairs and zonal spherical functions for finite groups. We refer [M95, VII.1] for the theory of finite groups. We continue to use the symbols in the previous §2. So H is the generic Hecke algebra associated to a finite Coxeter system (W, S) defined over the finite Galois extension K ⊃ C(us | s ∈ S). For a subset J ⊂ S, we call the subgroup WJ := hJi ⊂ W generated by J the parabolic subgroup, following the terminology in [GP00, 1.2]. The corresponding subalgebra

HJ := hTw | w ∈ WJ iK ⊂ H is called the parabolic subalgebra of H associated to J. We denote by eJ ∈ Z(HJ ) the idempotent (2.12) corresponding to the index representation of HJ . We regard it as an element of H. Thus we have 1 eJ = Tw ∈ H, PJ w WJ X∈ N where PJ := PWJ ∈ [us | s ∈ J] ⊂ K denotes the Poincar´epolynomial (2.10) of WJ . Now we consider the H-module eJ H. It is isomorphic to the induced module H H IndHJ (eJ HJ )= eJ HJ ⊗ J H

of the index representation eJ HJ . Since eJ is an idempotent, we have a K-algebra isomorphism

EndH(eJ H) −−→∼ eJ HeJ , ϕ 7−→ ϕ(eJ ). By [GP00, 9.1.9], the multiplicities m(V ) of the decomposition m(V ) eJ H = V ⊕ V Irr(H) ∈M into finite-dimensional simple H-modules V are preserved under the specialization H → Hq=1, us 7→ 1. In other words, the multiplicity m(V ) is the same as that in the irreducible decomposition of the W representation 1WJ of the finite group W obtained by inducing the trivial one-dimensional representation of the subgroup WJ . Now let us recall the notion of for finite groups [M95, VII.1, (1.1)]: A pair (G, K) of a finite group G and its subgroup K is called a Gelfand pair of finite groups if the induced G representation 1K is multiplicity-free, i.e., each of the non-zero multiplicities is one. Thus the following definition seems to be natural. Definition 3.1. Let (W, S) be a finite Coxeter system and J ⊂ S be a subset. The pair (S,J) or (H, HJ ) is called a Gelfand pair if the H-module eJ H is multiplicity-free, i.e., the pair (WS, WJ ) is a Gelfand pair of finite groups. Hereafter we assume (S,J) is a Gelfand pair, and denote by r eJ H = Vi i=1 M the decomposition into simple H-modules. Let χi be the character of Vi and χi∗ ∈ H be the element given by (2.7). Then we introduce:

Definition 3.2. For i =1, 2,...,r, we define an element ωi ∈ H by

ωi := χi∗eJ = eJ χi∗,

where the equality follows from χi∗ ∈ Z(H), explained in the paragraph of (2.13). We call ωi’s the zonal spherical functions of the Gelfand pair (S,J).

The zonal spherical functions ωi satisfy similar properties as the zonal spherical functions in the standard sense. The following lemma is a “Hecke-analogue” of [M95, VII.1, (1.4)].

Theorem 3.3. The elements ωi (i =1, 2,...,r) satisfy the following properties. (1) Let us express the expansion of ωi in terms of the basis {Tw | w ∈ W } as ωi = w W ωi(w)Tw. ∈ Then ωi(e) = 1. P (2) ωiωj = δi,j ciωi, where ci := cVi ∈ K is the Schur element associated to Vi (see §2.2). (3) ωi ∈ eJ HeJ ∩ Vi. (4) Zonal spherical functions ωi, i =1,...,r, form a K-basis of eJ HeJ . 6 1 Proof. − (1) By definition we have ωi = PJ v WJ Tv w W χi(Tw)Tw∨ . Since τ(TvTw∨) = δv,w 1 ∈ ∈ by (2.6), we have ωi(e) = P − χi(Tv). Since the module Vi lies in the induced module J v WJ P  P  H ∈ eJ on which Tv acts by the index representation (2.1), we have v WJ χi(Tv)= v WJ av = P ∈ ∈ PJ . Thus we have the result. P P (2) Denote by ei := eVi ∈ Z(H) the idempotent for Vi. By(2.13) we have χi∗ = ciei. Then the idempotence of eJ and (2.14) yields the conclusion. (3) We continue to use the symbol ei. Since eiH ≃ Vi as H-modules, we have ωi = cieiχi∗ ∈ Vi. (2) 2 yields ciωi = eJ (χi∗) eJ , and since ci is non-zero as explained in the paragraph of (2.11), we have ωi ∈ eJ HeJ . Thus we have the statement. (4) It follows from (3) that ωi’s are linearly independent in eJ H. On the other hand, Schur’s lemma yields dimK eJ HeJ = dimK EndH(eJ H)= r. Thus we have the conclusion. 

For the zonal spherical function ωi ∈ H and w ∈ W , we define ωi(Tw) ∈ K by the expansion

ωi = ωi(Tw)Tw∨. w W X∈ The limit ωi(Tw)|q=1 coincides with the value of the zonal spherical function on (W, WJ ) at the double coset WJ wWJ ∈ WJ \W/WJ . Unfortunately, ωi(Tw) is not constant on the double coset WJ wWJ , and at this point we have a discrepancy between Hecke- and classical settings. As a closing remark, let us recall that there are some explicit formulas of the values of zonal spherical functions for the Gelfand pairs of Sn and its , and for the pairs of GL(n, Fq) and its parabolic subgroups, in terms of terminating hypergeometric and basic hypergeometric series, respectively. See [Du79] and [M95, Chap. VII] for the detail. However, in the case of our zonal spherical functions for Hecke algebras, we don’t have such nice formulas at this moment. We invite the reader to tackle this problem.

Acknowledgements. The author thanks M. Hayashi and A. Hora for the fascinating collaboration in [HHY]. He also thanks Masatoshi Noumi for valuable comments on “quantized” spherical functions and hypergeometric functions.

References [Du79] C. F. Dunkl, Orthogonal functions on some permutation groups, in Relations between combinatorics and other parts of , 129–147, Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979. [HHY] M. Hayashi, A. Hora, S. Yanagida, Asymmetry of tensor product of asymmetric and invariant vectors arising from Schur-Weyl duality based on hypergeometric orthogonal polynomial, preprint, arXiv:2104.12635. [GP00] M. Geck, G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, 21. The Clarendon Press, Oxford University Press, New York, 2000. [J86] M. Jimbo, A q-analogue of U(gl(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys., 11 (1986), no. 3, 247–252. [M95] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, 1995.

Graduate School of Mathematics, Nagoya University Furocho, Chikusaku, Nagoya, 464-8602, Japan. Email address: [email protected]

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