arXiv:2104.12966v1 [math.RT] 27 Apr 2021 fHh oyoil [ polynomials Hahn of rga Elpi lers etxoeaosadln inva link and operators vertex algebras, “Elliptic Program itiuinhsfu o-eaieitgrparameters integer non-negative four has distribution otepriin( partition the to efn ar( pair Gelfand .. h esrsaeapaigi h lsia cu-eldaiyo duality Schur-Weyl classical the group in symmetric appearing the and space tensor the i.e., ytentrlato fta on that of action natural the by ubro onsa leof clue of a as points of number [ fGL( of eoe by denoted h ar(GL( pair the where r natural are [ formulas summation a troduced [ obtained also are formulas summation hypergeometric some a is which 2.2.2], Theorem SU(2)- the of p decomposition that [ of of Iwahori-Hecke D parts for Appendix the functions as explain spherical briefly included of us be theory to the planned of originally analogue was it and Hora, terminating se cee oevr h uuaiedsrbto function distribution cumulative the Moreover, scheme. Askey spher zonal of values as computation the ( in appear and polynomials, n [ and HHY S S hswr sspotdb upre yJP AEH rn Num Grant KAKENHI JSPS by supported by supported is work This hssotnt sasi ffo h olbrto ok[ work collaboration the of off spin a is note short This h function The n[ In e od n phrases. and words Key Date natural have arguments These n , n S n EFN AR N PEIA UCIN O IWAHORI-HECKE FOR FUNCTIONS SPHERICAL AND PAIRS GELFAND pedxC,isedo osdrn the considering of instead C], Appendix , HHY n( on ] n, F 2021.04.25. : m q q algebras. Abstract. + 1 = F × eoe h nt edo order of field finite the denotes C q q ,acrandsrt rbblt itiuini nrdcd hc origin which introduced, is distribution probability discrete certain a ], ihboksizes block with ) S 2 p q aaou of -analogue 4 ) ( aaous h etn saltl i rica ntevepito repre of viewpoint the in artificial bit little a is setting the -analogues, n F n, ⊗ x S − q 3 = ) n hpremti eis[ series -hypergeometric m n F + ecnie h cino h hvle ru GL( group Chevalley the of action the consider we , p , q .I diin h function the addition, In ). n ( S ) · · · eitouetento fGladpisadznlspherical zonal and pairs Gelfand of notion the introduce We x p P , − a upiigyrc rpry o xml,i a ewitna s a as written be can it example, For property. rich surprisingly has ) m ( x HHY + ( ,x x, × | ,n m, HHY ,m ,l k, m, n, q whr-ek lers efn ar,znlshrclfu spherical zonal pairs, Gelfand algebras, Iwahori-Hecke S n p S ,adby and ), − ( n oolre .. n ..] lhuhw a a httoecomputat those that say may we Although C.3.4]. and C.3.2 Corollaries , x 1 n − − S [ ) hoe ..] anplnmasare polynomials Hahn 2.2.1]. Theorem , 4 eew eoe by denoted we Here . m sthe is m q F n m, HHY ste elcdby replaced then is ) dfrain sfrtegopato,isedo h permutatio the of instead action, group the for As -deformation. HHY 3 -bimodule hpremti rhgnlplnma itn ntefis ieo th of line first the in sitting polynomial orthogonal -hypergeometric for ) F and F q n q ( q h sg fHh oyoil sznlshrclfntosfrth for functions spherical zonal as polynomials Hahn of usage The . aaous hc r icse n[ in discussed are which -analogues, C q C21),adaeotie h orsodn ai hypergeomet basic corresponding the obtained are and (C.2.11)], , ) where )), hc ocr hsnote. this concern which ] itgr nohrwrs etk h formula the take we words, other In -integer. U 2 n x ) ( − ⊗ n 0 = HHY − n HNAO YANAGIDA SHINTAROU m ( x,x C = q 1. , . , 2 1 ) ALGEBRAS ⌊ ) hoe ..] ntecus fdrvn hs theorems, these deriving of course the In 2.2.4]. Theorem , P , . . . , x X n/ P introduction P h orsodn reuil U2-ersnain The SU(2)-representation. irreducible corresponding the p 1 =0 1 ( ( 2 ( ( n F x F ⌋ ,n m, tno pc ( space -tensor q n a ewitnb igeRchplnma [ polynomial Racah single a by written be can ) riant”. q n U ⌊ ) eoe h rjcieln o h ierspace linear the for line projective the denotes ) V ( n/ 1 ( = n q ( n − Hh oyoil sznlshrclfntosfor functions spherical zonal as polynomials -Hahn − 2 − x,x ⌋ C x,x m, ,m ,l k, m, n, . 2 ) ) ) ⊠ [ F n h irreducible the q HHY ] V q eoe h aia aaoi subgroup parabolic maximal the denotes ) HHY , ( n e 9039 n lob h SSBilateral JSPS the by also and 19K03399, ber 1.1 − lers oepantemtvto,let motivation, the explain To algebras. n h rbblt asfnto is function mass probability the and , x,x ihMsht aah n Akihito and Hayashi Masahito with ] P oolre .. n 2.2.5]. and 2.2.3 Corollaries , ,w consider we ), n, ) u x , clfntosfrteGladpair Gelfand the for functions ical =0 h pca ntr ru SU(2) group unitary special the f F HHY q ucin o Iwahori-Hecke for functions n( on ) 3 nctions. p F ( u 2 S hpremti orthogonal -hypergeometric a lob rte ya by written be also can ) pr h i st iean give to is aim The aper. pedxC.Teei in- is There C]. Appendix , n C rpeetto associated -representation 2 ) P tsi h irreducible the in ates
1 P ( etto hoy In theory. sentation F 1 q n ( ) F hc sinduced is which q n )
[ = 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