FREE PROBABILITY ON HECKE ALGEBRAS AND CERTAIN GROUP C -ALGEBRAS INDUCED BY HECKE ALGEBRAS. Ilwoo Cho

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1 Opuscula Math. 36, no. 2 (2016), Opuscula Mathematica FREE PROBABILITY ON HECE ALGEBRAS AND CERTAIN GROUP C -ALGEBRAS INDUCED BY HECE ALGEBRAS Ilwoo Cho Communicated by P.A. Cojuhari Abstract. In this paper, by establishing free-probabilistic models on the Hecke algebras H pgl 2pQ pqq induced by p-adic number fields Q p, we construct free probability spaces for all primes p. Hilbert-space representations are induced by such free-probabilistic structures. We study C -algebras induced by certain partial isometries realized under the representations. eywords: free probability, free moments, free cumulants, Hecke algebras, normal Hecke subalgebras, representations, groups, group C -algebras. Mathematics Subject Classification: 05E15, 11R47, 46L54, 47L15, 47L INTRODUCTION In this paper we study free-probabilistic models for Hecke algebras and study representations under the models, and investigate groups generated by certain operators under the representations. In [7], the author and Gillespie considered certain embedded free-probabilistic subalgebras of Hecke algebras induced by p-adic number fields for primes p. And, in [2], the author extended the free-probabilistic representations of [7] to those fully on the given Hecke algebras, and investigated elements of Hecke algebras as operators realized under the representations. Especially, the spectral theory of such Hilbert-space operators was considered in [2]. As a continuation, here, we keep studying free probability on the Hecke algebras in the extended sense of [2], and concentrate on studying certain group C -(sub-)algebras determined by the representations (under quotient) BACGROUND We have considered how primes (or prime numbers) act on operator algebras. The relations between primes and operator algebra theory have been studied from various c AGH University of Science and Technology Press, rakow

2 154 Ilwoo Cho different approaches. For instance, in [1], we studied how primes act on certain von Neumann algebras generated by p -adic and Adelic measure spaces. Also, the primes as operators in certain von Neumann algebras, have been studied in [3] and [5]. Independently in [6] and [4] we have studied primes as linear functionals acting on arithmetic functions, i.e., each prime p induces a free-probabilistic structure pa, g p q on the algebra A of all arithmetic functions. In such a case, one can understand arithmetic functions as rein-space operators (for fixed primes) via certain representations (see [8]). These studies are motivated by number-theoretic results (e.g., [9, 10] and [14]) under free probability techniques (e.g., [11, 12] and [13]) MOTIVATION In modern number theory and its applications, p-adic analysis provides important tools not only for studying mathematical analysis, analytic number theory and non-archimedian analysis (e.g., [1, 3, 7, 9] and [10]), but also for studying geometry at small distances in mathematical quantum physics (e.g., [14]). So, it is interested in both various mathematical fields and related scientific fields. In [2] we studied free probability on Hecke algebras (see Sections 3 and 4 below). From the free-probabilistic models on Hecke algebras, we established certain representations of Hecke algebras, and considered corresponding C -algebras of Hecke algebras obtained from the representations, i.e., we understand every Hecke-algebra element as a Hilbert-space operator. Especially in [2], spectral properties (self-adjointness, normality, isometry-property, unitarity, etc.) of such operators were characterized. In this paper we are typically interested in projections and partial isometries induced by generating elements of HpG p q. By understanding them pure operator-theoretically we construct group C -algebras generated by certain nice partial isometries having their common initial-and-final projections. The operator-algebraic properties of such C -algebras will be studied as embedded C -subalgebras of the C -algebra induced by Hecke algebras. Our study will provide bridges among number theory, operator algebra, operator theory and free probability OVERVIEW In Section 2 we introduce definitions and fundamental properties for our work. In Sections 3 and 4 we briefly review our free probability models on Hecke algebras. Some free-moment and free-cumulant computations are provided for our main results. In Section 5 we establish Hilbert-space representations of Hecke algebras and construct corresponding C -algebras, as operator-algebraic structures containing full free-probabilistic information of Hecke algebras. In Section 6 we study partial isometries and projections induced by generating elements of Hecke algebras under our representations in detail. Projections and partial isometries in our Hecke C -algebras have been considered in [2], but we here provide much more detailed properties and characterizations of them (Theorem 6.1 and Theorem 6.2) independently. Moreover, we fix finitely many partial isometries,

3 Free probability on Hecke algebras and certain group C -algebras having identical initial-and-final projections, and then construct groups generated by such partial isometries, as multiplicative subgroups of Hecke C -algebras. We study isomorphism theorems of such groups (see Theorem 6.3). Naturally, corresponding group C -algebras will be constructed as embedded C -subalgebras of the Hecke C -algebras. We consider structure theorems of such group C -algebras in Theorem 6.4 and Corollary 6.5. In Section 7 free probability on these group C -algebras will be studied. We study free-distributional data of operators in the algebras by computing free-moments (Theorem 7.1 and Corollary 7.2), and consider freeness conditions (Theorem 7.6) on the group C -algebras by observing free-cumulants (Theorem 7.4) of generating operators. 2. DEFINITIONS AND BACGROUND In this section we review concepts and backgrounds of our proceeding works THE HECE ALGEBRA OVER GL 2 pq p q Throughout this section let p be a fixed prime, and let Q p be the p-adic number field for p. This set Q p is by definition the completion of the rational numbers Q with respect to the p-adic norm ˇ q p ˇp k a ˇ b k ˆ1 p for q p k a b P Q and k P Z. Define now the (multiplicative) group GL 2 pq p q of all invertible p2 ˆ 2q-matrices over the p-adic number field Q p, "ˆa b GL 2 pq p q P M c d 2 pq p q ˇ a, b, c, d P Q * p,, ad bc 0 where M 2 pq p q means the set of all p2 ˆ 2q-matrices over Q p. In the rest of this paper we denote GL 2 pq p q simply by G p, if there is no confusion. The group G p is locally profinite coming from the topology on Q p, i.e., it has a neighborhood base of the identity u p of G p, consisting of the compact-open subgroups k u p ` pp k qgl 2 pz p q for all k P N, where GL 2 pz p q means the subset of GL 2 pq p q whose elements have their entries in Z p, and where ˆ1 0 u p is the identity matrix of M pq p q. Then the subgroup 0 GL 2 pz p q forms the maximal compact-open subgroup of G p.

4 156 Ilwoo Cho Now let pv, πq be a representation of G p, that is V is a vector space, and π is a group action, π : G p Ñ GLpV q acting on V, where GLpV q is the set of all invertible linear transformations on V. Definition 2.1. We say a representation pv, πq is a smooth representation, if given any vector v P V, there is a compact-open subgroup of G p, such that πpyqv v for all y P. Denote by V the set of vectors in V that are fixed by under the action of π. Then the definition of smoothness implies that ď V V. ĎG p: compact-open Given two smooth representations pv 1, π 1 q and pv 2, π 2 q of G p, we denote by the set of C-linear maps Hom Gp pπ 1, π 2 q, f : V 1 Ñ V 2 such that f π 1 pgq π 2 pgq f for all g P G p. Definition 2.2. Define the Hecke algebra HpG p q of G p by HpG p q tf : G p Ñ C f has compact-open support, and it is ρ-smoothu. (2.1) The ρ-smoothness means that HpG p q is a smooth representation of G p under right translation. In other words, for any element f P HpG p q, there is a compact-open subgroup of G p such that ρpyqfpgq fpgyq fpgq (2.2) for all g P G p. We sometimes say also that f is locally constant. We make HpG p q into an associative algebra by taking f 1, f 2 P HpG p q and defining convolution (as a vector multiplication) ż pf 1 f 2 q pgq f 1 pxqf px 1 2 gqdµ p pxq, (2.3) G p where µ p denotes a left Haar measure on the locally compact-open group G p.

5 Free probability on Hecke algebras and certain group C -algebras FREE PROBABILITY Throughout this paper we use Speicher s combinatorial free probability techniques in the sense of [12] (also, see cited papers therein). The original analytic free probability theory is established by Voiculescu, and since the mid 1980 s, it has developed as one of the main branches of operator algebra theory. By replacing independence of classical probability theory to (noncommutative) freeness, we can have the noncommutative (and hence, possibly commutative) operator-algebraic and operator-theoretic probability and corresponding statistics (for instance, free stochastic calculus, etc). Such a noncommutative(-or-commutative)-algebraic extended probability theory, called free probability, has various applications not only in mathematics (operator theory, in particular, spectral theory, and operator algebra, see e.g. [11]), but also in related scientific fields (e.g., free entropy theory, quantum probability, and quantum statistics, etc). In combinatorial free probability the free-probabilistic information of given operators in an algebra is determined by free moments or free cumulants (see e.g., [12]). In fact free moments and free cumulants are equivalent under the Möbius inversion; but free moments are used for studying free-distributional data of operators, while free cumulants are used for studying freeness among operators in the algebra. We refer readers to [12] and [13] for more about free probability theory. Especially, we will use the same concepts and results of [12] in this paper (without introducing them precisely) GROUP ALGEBRAS Let G be a countable discrete group. Then one can construct the algebra A G by # + ÿ A G CrGs t g g : t g P C for all g P G, gpg where ř means a finite sum, i.e., A G is the algebra generated by G. We call A G, the group algebra generated by G. Each group algebra A G is understood as a -algebra over C, by defining the adjoint ( ) on it by ÿ t g g gpg def ÿ gpg t g g 1, where g 1 in the right-hand side mean group-inverse of g. All groups G of this paper are assumed to be countable discrete groups. Every group algebra A G acts on the Hilbert space H G l 2 pgq via a group-action u, under the left regular unitary representation denoted by ph G, uq, where l 2 pgq means the l 2 -space with its orthonormal basis (or its Hilbert basis) tξ g : g P Gzte G uu,

6 158 Ilwoo Cho where e G is the group-identity of G, satisfying xξ g1, ξ g2 y 2 δ g1,g 2, where x, y 2 means the inner product on H G and δ means the ronecker delta. In particular, the group-action u acts as follows: for each g P G, the image upgq, denoted by u g, becomes a unitary operator in the sense that: u g u 1 g, where u g means the (Hilbert-space-operator-)adjoint of u g, and u 1 g means the (operator-)inverse of u g on H G. In particular, the unitary operators tu g u gpg satisfy for all g 1, g 2 P G, and ξ g2 P H G, and and u g1 pξ g2 q def ξ g1 ξ g2 ξ g1g 2 u g1 u g2 u g1g 2 for all g 1, g 2 P G, u g u 1 g u g 1 for all g P G, where u 1 g mean the operator-inverses of u g for all g P G. By construction it is easy to check that a group algebra A G is a ( -)subalgebra of the operator algebra BpH G q, consisting of (bounded linear) operators on H G (pure algebraically, without considering topology). So under operator-norm topology of BpH G q, we can have the group C -algebra A G ; also, under weak-operator topology, one can have the group von Neumann algebra (or the group W -algebra) w A G, etc. Let A G be the group algebra. Define a linear functional by tr G ÿ tr G : A G Ñ C gpg t g g def t eg. Then it is a well-defined linear functional. Moreover, it satisfies tr G px 1 x 2 q tr G px 2 x 1 q for all x 1, x 2 P A G, even though x 1 x 2 x 2 x 1 in A G, i.e., tr G is a trace on A G. We usually call tr G the canonical trace on A G (e.g., [11]). Thus, the pair pa G, tr G q forms a free probability space in the sense of Section 2.2. This free probability space pa G, tr G q is called the (canonical) group(-algebra)free probability space (under topologies, the group C -free probability space, or the group W -probability space, etc). 3. NORMAL HECE PROBABILITY SPACES In this section we review free-probabilistic structures obtained in [7], and main results of [7] will be introduced for our future work.

7 Free probability on Hecke algebras and certain group C -algebras NORMAL HECE SUBALGEBRAS H Yp OF HpG p q Notice, first that, by the very definition (2.1), the Hecke algebra HpG p q can be re-defined by HpG p q C! f Nÿ ˇ t j χ xjˇn P N, and t j P C, is a compact-open subgroup of G p, depending on f )ı for all x j P G p, j 1,..., N, (3.1) where C rxs mean algebras generated by X under the usual functional addition and convolution in the sense of Section 2.1, and χ Y mean characteristic functions of µ p -measurable subsets Y of G p, where µ p is in the sense of (2.2). The set X p! f Nÿ ˇ t j χ xjˇn P N, and t j P C, is a compact-open subgroup of G p, depending on f ) for all x j P G p, j 1,..., N (3.2) generating the Hecke algebra HpG p q, is said to be the generating set of HpG p q, and we call elements of X p of (3.2) generating elements of HpG p q, i.e., By (3.1) and (3.3), one may write HpG p q C rx p s. (3.3)! ÿ N HpG p q t j χ xj j ˇˇˇN P N, and tj P C, and j are compact-open subgroups of G p, ) for all x j P G p, j 1,..., N, (3.4) set-theoretically. By construction HpG p q is a well-defined vector space over C. As in Section 2.1, the convolution ( ) on HpG p q, as a vector multiplication, is defined by ż pf 1 f 2 qpgq f 1 pxqf px 1 2 gqdµ p pgq G p for all f 1, f 2 P HpG p q, for all g P G p. Proposition 3.1 ([7]). Let χ x1 1, χ x2 2 be generating elements of HpG p q, for x j P G p, and compact-open subgroups j of G p for j 1, 2. Then pχ x1 1 χ x2 2 q pgq µ p `x1 1 X g x (3.5) for all g P G p.

8 160 Ilwoo Cho Thus by (3.5), we obtain the following general result; if f j ř n j k1 t j,kχ xj,k j are generating elements of HpG p q in X p, for j 1, 2, then pf 1 f 2 qpgq ÿn 1 k1 l1 ÿn 2 pt 1,k t 2,l q µ p x 1,k 1 X g x 1 2 for all g P G p. Without loss of generality, for any x P G p, one can understand 2,l χ x pgq µ ppx X gq µ p pxq µ ppx X gq µ p pq (3.6) by (2.2). We now consider specific generating elements χ x in X p, where are normal compact-open subgroups of G p. Recall that a subgroup is normal in an arbitrary group Γ, if g g for all g P Γ. As usual, we denote this normal subgroup-inclusion by Γ. Define a subset Y p of the generating set X p of HpG p q by Then we have a subalgebra Y p def # + ÿ N t j χ xj P X p G p. (3.7) H Yp def C ry p s of HpG p q. (3.8) Proposition 3.2 ([7]). Let χ xj j for j 1, 2. Then P H Yp, where x j P G p, and j G p compact-open, χ x1 1 χ x2 2 µ p p 1 X 2 qχ x1x 2 1 2, (3.9) where 1 2 is the product group of 1 and 2 in G p. Definition 3.3. Let Y p be the subset (3.7) of the generating set X p, and let H Yp C ry p s be the subalgebra (3.8) of the Hecke algebra HpG p q. Then we call Y p and H Yp, the normal sub-generating set of X p, and the normal Hecke subalgebra of HpG p q, respectively. For convenience, denote ś N x j and Ṋ j simply by x 1,...,N and 1,...,N, respectively, for all N P N, where x 1,..., x N P G p and 1,..., N are (normal) compact-open subgroups of G p. Also, denote for all N P Nzt1u. 1,...,pN 1q X N by o 1,...,N

9 Free probability on Hecke algebras and certain group C -algebras We obtain the following general computations. Proposition 3.4. Let χ xj j for j P N. Then H Yp be generating elements of the normal Hecke subalgebra for all N P N. N χ xj j `µ p p1,2qµ o p p1,2,3q o... µ p p1,...,n q o χ x1,...,n 1,...,N (3.10) Proof. The proof of (3.12) is done by (3.9), inductively (e.g., [2] and [7]). From now on, let us denote the convolution f... f of n-copies of f simply by f pnq for all n P N and f P HpG p q FREE-PROBABILISTIC MODELS ON H Yp Let HpG p q be the Hecke algebra generated by the generalized linear group G p GL 2 pq p q over the p-adic number field Q p, for a fixed prime p. From Section 3.1, we start to understand this algebra HpG p q as an algebra C rx p s generated by X p of (3.1), consisting of C-valued functions f formed by f Nÿ t j χ xj for t j P C, x j P G p, (3.11) where is a compact-open subgroup of G p, for N P N. So, to consider free-distributional data, we concentrate on generating elements χ x s and e x s, for x P G p, and compact-open subgroups. Moreover, in this section, we restrict further our interests to the normal Hecke subalgebra H Yp of HpG p q, for a fixed prime p. Let u p be the group-identity of G p, i.e., u p ˆ1 0 P G 0 1 p GL 2 pq p q. For the fixed u p define now a linear functional ϕ p on H Yp by ϕ p pfq def fpu p q for all f P H Yp. (3.12) The construction of the linear functional ϕ p on H Yp (originally introduced in [7]) is motivated by the canonical traces on group von Neumann algebras (e.g., [11]), and thepoint-evaluation linear functionals on arithmetic functions in the sense of [4 6] and [8]. Clearly, the morphism ϕ p is a well-defined linear functional on H Yp, and hence, the pair ph Yp, ϕ p q forms a free probability space in the sense of Section 2.2. Definition 3.5. We call the linear functional ϕ p of (3.12) on the normal Hecke subalgebra H Yp, the canonical linear functional. And the corresponding free probability space ph Yp, ϕ p q is said to be the normal Hecke probability space.

10 162 Ilwoo Cho Then we obtain the following fundamental free-moment computations. Proposition 3.6 ([7]). Let χ x, χ xj j, e x, e xj j be generating free random variables in the normal Hecke probability space ph Yp, ϕ p q for all j P N. Then ˆ ϕ p N χ xj j µ pp1,2q o... µ p p1,...,n o qµ ppx 1,...,N 1,...,N X 1,...,N q (3.13) µ p p 1,...,N q for all N P N. Indeed, ˆ ϕ p N χ xj j ϕ p `µp p1,2qµ o p p1,2,3q o... µ p p1,...,n o qχ x1,...,n 1,...,N by (3.10) µ p p o 1,2qµ p p o 1,2,3q... µ p p o 1,...,N qϕ p `χx1,...,n 1,...,N µ p p o 1,2qµ p p o 1,2,3q... µ p p o 1,...,N qχ x1,...,n 1,...,N pu p q by (3.12) µ p p o 1,2qµ p p o 1,2,3q... µ p p o 1,...,N q µ ppx 1,...,N 1,...,N X 1,...,N q µ p p 1,...,N q by (3.6) µ pp o 1,2qµ p p o 1,2,3q... µ p p o 1,...,N qµ ppx 1,...,N 1,...,N X 1,...,N q µ p p 1,...,N q for all N P N. Let χ x1 1,..., χ xn N P ph Yp, ϕ p q for N P N. Then k p N pχ x 1 1,..., χ xn N q ÿ ź ˆ ϕ p πpncpnq V Pπ by the Möbius inversion of Section 2.2 by (3.13), where ÿ πpncpnq ź jpv χ x ij ij V pi 1,...,i V qpπ µ `0 V, 1 V pµ p pv qq µ `0 V, 1 V, µ p pv q µ ppi o 1,i 2 q... µ p pi o 1,...,i V qµ p px i1,...,i V i1,...,i V X i1,...,i V q, µ p p i1,...,i V q (3.14)

11 Free probability on Hecke algebras and certain group C -algebras are the block-depending free moments for all V P π and π P NCpNq, where k p np...q means free cumulant determined by ϕ p as in Section 2.2. By (3.14) one can get the following freeness condition (3.15) on the normal Hecke subalgebra H Yp. And this freeness condition shows that classical independence guarantees our freeness. Proposition 3.7 ([7]). Let f j χ j be free random variables in the normal Hecke free probability space ph Yp, ϕ p q for j 1, 2. Then f 1 and f 2 are free in ph Yp, ϕ p q ô µ p p o 1,2q µ p p 1 qµ p p 2 q. (3.15) 4. FREE PROBABILITY ON HpG p q In this section we extend the free probability on the normal Hecke subalgebra H Yp of Section 3.2 to free probability fully on the Hecke algebra HpG p q. For more information about such extensions, see [2]. Let G be an arbitrary group and let be a subgroup of G. The normal core Core G pq of in G is defined by the subgroup of G, Core G pq def X gpg `g 1 g. (4.1) Then the normal core Core G pq is the maximal normal subgroup of G contained in, i.e., Core G pq G and Core G pq ď. (4.2) For convenience, we denote the normal core Core G pq of (4.1) satisfying (4.2) simply by G. Define now a linear transformation E p on the Hecke algebra HpG p q by a morphism satisfying (4.3) and (4.4) below: E p pχ x q # χ xgp if x x, 0 HpGpq otherwise (4.3) and # µ p p o E p pχ x1 1 χ x2 2 q 1,2qχ x1,2 1,2:Gp if x i j j x j for all i, j P t1, 2u, 0 HpGpq otherwise, (4.4) where Gp and 1,2:Gp mean the normal cores of and 1,2 in G p, respectively, and where 0 HpGpq is the zero element of HpG p q. By (4.3) and (4.4), if j are compact-open subgroups of G p, and x i P G p, and if x i j j x i for all i, j 1,..., N, (4.5)

12 164 Ilwoo Cho for N P N, then E p pχ x χ xn N q E p `µp p o 1,2qµ p p o 1,2,3q... µ p p o 1,...,N q χ x1,...,n 1,...,N (4.6) µ p p o 1,2qµ p p o 1,2,3q... µ p p o 1,...,N qχ x1,...,n 1,...,N:Gp (4.7) inductively by (4.4). Remark that if the condition (4.5) holds, then the formula N χ xj j µ p p1,2qµ o p p1,2,3q o... µ p p1,...,n o qχ x1,...,n 1,...,N (4.8) holds in HpG p q, without normality of 1,..., N in G p (see [2]), and hence, the formula (4.6) holds, and hence the equality (4.7) holds, by (4.3) and (4.6). Proposition 4.1. Let f j χ xj j be generating elements of the Hecke algebra HpG p q, for j 1,..., N, for N P N, and let E p be the linear transformation (4.4) on HpG p q. If x i j j x i for all i, j 1,..., N, then ˆ E p N f j Nź µ p p1,...,jq o χ x1,...,n 1,...,N:Gp. (4.9) j2 Otherwise, they are identical to the zero element 0 HpGpq of the Hecke algebra HpG p q. Proof. The proof of (4.9) is done by (4.5) and (4.8). See [2] for more details. By construction it is not difficult to check that the linear transformation E p maps HpG p q onto the normal Hecke subalgebra H Yp. Moreover, this morphism E p is idempotent in the sense that E 2 ppfq E p pe p pfqq E p pfq for all f P HpG p q, because normal cores are normal subgroups of G p. Definition 4.2. We will call the morphism E p of (4.2), the normal-coring on HpG p q. Define now a linear functional ψ p on the Hecke algebra HpG p q by ψ p def ϕ p E p on HpG p q. (4.10) By the linearity of both the canonical linear functional ϕ p on H Yp and the normal-coring E p on HpG p q, the morphism ψ p is a linear functional on HpG p q. We call the linear functional ψ p of (4.10), the normal-cored (canonical) linear functional on HpG p q. So, the pair phpg p q, ψ p q forms a free probability space. Definition 4.3. The free probability space phpg p q, ψ p q of the Hecke algebra HpG p q and the normal-cored linear functional ψ p of (4.10) is said to be the normal-cored Hecke probability space.

13 Free probability on Hecke algebras and certain group C -algebras Generally we obtain the following joint free-moment computations. Theorem 4.4. Let phpg p q, ψ p q be the normal-cored Hecke probability space, and let f j χ xj j be generating free random variables in phpg p q, ψ p q for j P N. If the condition (4.5) holds for N P N, then we obtain ψ p ˆ N f j `µp p o 1,2qµ p p o 1,2,3q... µ p p o 1,...,N q µ p `x1,...,n 1,...,N:G X 1,...,N:Gp µ p `1,...,N:Gp (4.11) for all N P N, where 1,...,N:Gp is in the sense of (4.2). If there exists at least one pair pi, jq P t1,..., Nu 2, for N P N, such that x i j j x i in G p, then the formulas (4.11) vanish in HpG p q. Proof. Suppose first that x i j j x i for all i, j 1,..., N, for N P N, ˆi.e., assume that the condition (4.5) holds. Then we have ψ p N f j ψ p `µp p1,2qµ o p p1,2,3q o... µ p p1,...,n o q χ x 1,...,N 1,...,N by (4.6) µ p p1,2qµ o p p1,2,3q o... µ p p1,...,n o qψ p `χx1,...,n 1,...,N µ p p1,2q o... µ p p1,...,n o q ϕ p `Ep pχ q x1,...,n 1,...,N µ p p1,2q o... µ p p1,...,n o q ϕ p χ x1,...,n 1,...,N:Gp ˆ µ p p1,2q o... µ p p1,...,n o q µ ppx 1,...,N 1,...,N:G X 1,...,N:Gp q µ pp 1,...,N:Gp q by (3.9) µppo 1,2 q...µppo 1,...,N qµppx 1,...,N 1,...,N:G X 1,...,N:Gp q. µ pp 1,...,N:Gp q So, the formula (4.11) holds. Of course if there exists at least one pair pi, jq, such that x i j j x i, then the formulas (4.11) and (4.12) simply vanish, by (4.3) and (4.4). So we obtain that ˆ ψ p N χ j µ pp1,2qµ o p p1,2,3q o... µ p p1,...,n o qµ pp 1,...,N:Gp q µ p p 1,...,N:Gp q µ p p o 1,2,qµ p p o 1,2,3,q... µ p p o 1,...,N q, (4.12) by (4.11). Now let 1 and 2 be compact-open subgroups of G p, and let χ j be corresponding free random variables in the normal-cored Hecke probability space phpg p q, ψ p q. Suppose k N p...q is the free cumulant for the normalized linear functional ψ p. Then,

14 166 Ilwoo Cho for any pi 1,..., i N q P t1, 2u N, for all N P N, we obtain the following free cumulant computation: k N χ i1,..., χ in ÿ ˆ Π µ ppv q µ `0 V, 1 V V Pπ (4.13) πpncpnq with µ p pv q µ p p o i j1,i j2 qµ p o i j1,i j2,i j3... µ p o i j1,...,i jk, by (4.12), whenever V pj 1,..., j k q P π for all π P NCpNq and for all N P N, where µ p pv q are the V -block-depending free moments. By the above joint free-cumulant formula (4.13), we obtain the following freeness condition on the normalized Hecke probability space phpg p q, ψ p q. Theorem 4.5 ([2]). Let f j χ j and h j e j be free random variables in the normal-cored Hecke probability space phpg p q, ψ p q for j 1, 2. Then f 1 and f 2 are free in phpg p q, ψ p q ô µ p p o 1,2q µ p p 1 qµ p p 2 q. (4.14) 5. REPRESENTATIONS ON NORMAL-CORED HECE PROBABILITY SPACES In this section we introduce representations of the normal-cored Hecke probability spaces phpg p q, ψ p q, for primes p. Let p be a fixed prime, and let phpg p q, ψ p q be the corresponding normal-cored Hecke probability space. Define a sesqui-linear form on the Hecke algebra HpG p q, by where r, s p : HpG p q ˆ HpG p q Ñ C rf 1, f 2 s p def ψ p pf 1 f 2 q for all f 1, f 2 P HpG p q, (5.1) def f pxq fpxq in C for all x P G p, where z means the conjugate of z for all z P C. We call the above unary operation f P HpG p q ÞÝÑ f P HpG p q, (5.2) the adjoint. And the element f of (5.2) is said to be the adjoint of f. Since the adjoint (5.2) is well-defined on HpG p q, one may understand our Hecke algebra HpG p q as a -algebra over C. The form r, s p of (5.1) is indeed sesqui-linear, since and for all f 1, f 2, f 3 P HpG p q and t 1, t 2, t 3 P C. rt 1 f 1 ` t 2 f 2, f 3 s p t 1 rf 1, f 3 s ` t 2 rf 2, f 3 s rf 1, t 2 f 2 ` t 3 f 3 s p t 2 rf 1, f 2 s p ` t 3 rf 1, f 3 s p

15 Free probability on Hecke algebras and certain group C -algebras Consider now that, for any fixed generating element χ x of HpG p q, for x P G p, and a compact-open subgroup of G p, we have rtχ x, tχ x s p ψ p `tχx tχ x t 2 ψ p pχ x χ x q by the sesqui-linearity of r, s p, where t means the modulus? t t of t, by (4.11), i.e., # t 2 ψ p pµ p pq χ x2 q if x x 0 otherwise $ & 2 ˆ µ ppx µ p pq t 2 Gp X Gp q µ pp Gp q if x x % 0 otherwise $ ˆ & t 2 µ ppq µ ppx 2 Gp X Gp q µ pp Gp q if x x % 0 otherwise ˆµp rtχ x, tχ x s p t 2 pqµ p px 2 Gp X Gp q, or 0, (5.3) µ p p Gp q where Gp is the normal core of in G p. So, by (5.3), we obtain that rtχ x, tχ x s p ě 0 (5.4) for all x P G p, for all compact-open subgroups of G p, for all t P C. By (5.4) one can get in general that rf, fs p ě 0 for all f P HpG p q. (5.5) Proposition 5.1 ([2]). The sesqui-linear form r, s p on the Hecke algebra HpG p q forms a pseudo-inner product on HpG p q. Suppose is a nonempty proper normal compact-open subgroup of G p and let x be the left coset of by x P G p. As non-empty subsets of G p, it is possible that In such a case we have x X, and hence, µ p px X q 0. rχ x, χ x s p ψ p pµ p pqχ x q ϕ p pµ p pq χ x q µ ppqµ p px X q µ p pq i.e., there exist nonzero elements f of HpG p q such that rf, fs p 0. µ p px X q 0,

16 168 Ilwoo Cho Indeed, if x x in G p, then, by the very definition of E p, and hence, even though χ x 0 HpGpq, i.e., E p pχ x χ x q 0 HpGpq, ψ p pχ x χ xq ϕ p `0HpGpq 0, Df 0 HpGpq : rf, fs p 0. (5.6) So the pseudo-inner product space phpg p q, r, s p q is not an inner product space, by (5.6). When we understand our Hecke algebra HpG p q as a pseudo-inner product space, we denote it by H p. On the pseudo-inner product space H p define a relation R p by f 1 R p f 2 def ðñ rf 1, f 1 s p rf 2, f 2 s p. (5.7) By the very definition (5.7) of R p, it is an equivalence relation on H p. Definition 5.2. Let H p be the pseudo-inner product space (5.6), and let R p be the equivalence relation (5.7) on H p. Define the quotient space H p by H p H p {R p, (5.8) equipped with the inherited pseudo-inner product, also denoted by r, s p on it. Then H p ph p, r, s p q ph p {R p, r, s p q is called the (normal-cored) Hecke inner product space. From now on, if there is no confusion we denote equivalence classes rfs Rp th P H p : hr p fu simply by f in the Hecke inner product space H p. Indeed, our Hecke inner product space H p is an inner product space, by R p of (5.7), i.e., it satisfies rf, fs p 0 ðñ f 0 Hp 0 Hp {R p, (5.9) where 0 Hp is the zero element of H p. For the given inner product space H p, one can define the corresponding norm } } p on H p by }f} p def and the corresponding metric d p on H p by b rf, fs p for all f P H p, (5.10) d p pf 1, f 2 q }f 1 f 2 } p for all f 1, f 2 P H p. (5.11)

17 Free probability on Hecke algebras and certain group C -algebras Definition 5.3. Construct the d p -metric topology closure of H p, also denoted by H p, where d p is in the sense of (5.11) induced by the norm } } p of (5.10). It is called the (normal-cored) Hecke Hilbert space. Then by the very construction of the Hecke Hilbert space H p from the normal-cored Hecke probability space phpg p q, ψ p q, the algebra HpG p q acts on H p via an algebra-action α p ; α p pfqphq f h for all h P H p, (5.12) for all f P HpG p q. More precisely, the above relation (5.12) means α p pfqphq α p pfq `rhs Rp rf hsrp (5.13) in H p for f P HpG p q. For convenience, we denote α p pfq by α p f for all f P HpG pq. The above morphism α p of (5.12) and (5.13) is indeed a well-defined algebra-action of HpG p q acting on H p, since α p f 1 f 2 phq f 1 f 2 h f 1 pf 2 hq f 1 α p f 2 phq α pf1 α p f 2 phq for all h P H p and f 1, f 2 P HpG p q, i.e., for all f 1, f 2 P HpG p q. Also, α p satisfies that ı α pf ph 1q, h 2 rf h 1, h 2 s p for all h 1, h 2 P H p and f P HpG p q, i.e., p α pf1 α pf2 phq α p f 1 f 2 α p f 1 α p f 2 on H p (5.14) ψ p ppf h 1 q h 2 q ψ p ph 1 f h 2 q ψ p ph 1 ph 2 fqq ψ p ph 1 pf h 2 q q ı rh 1, f h 2 s p h 1, α p f ph 2 q α p f α p f on H p for all f P HpG p q. (5.15) Therefore, the morphism α p of (5.12) is a -algebra-action of HpG p q acting on H p, by (5.14) and (5.15). Theorem 5.4. The pair ph p, α p q of the Hecke Hilbert space H p and the morphism α p of (5.12) forms a Hilbert-space representation of the Hecke algebra HpG p q acting on H p. Proof. The proof is done by (5.13), (5.14) and (5.15). (See [2] for more details.) p

18 170 Ilwoo Cho We call the algebra-action α p of (5.12) the (normal-cored) Hecke(-algebra) action of HpG p q acting on H p. Definition 5.5. The Hilbert-space representation ph p, α p q of the Hecke algebra HpG p q is called the (normal-cored) Hecke representation (of the normal-cored Hecke probability space phpg p q, ψ p q). 6. CERTAIN PROJECTIONS AND PARTIAL ISOMETRIES ON H p In this section under the Hecke representation ph p, α p q of the Hecke probability space phpg p q, ψ p q, certain generating elements of HpG p q will be considered as Hilbert-space operators on H p (under quotient). In particular, we are interested in partial isometries induced by generating elements and their initial and final projections. Already in [2] we studied some operator-theoretic information; self-adjointness, normality, unitarity, isometry-property and hyponormality; of such operators. In particular, we realized that, by the very constructions of the Hecke algebra HpG p q and our representation ph p, α p q, there are no isometries (and hence, no unitaries) formed by α p tχ x, for t P C, x P G p, and compact-open subgroups of G p. However, operators α p tχ x are always normal on H p. Since there are neither isometries nor unitaries we are interested in the operators α p tχ x which are projections, and partial isometries having their identical initial-and-final projections on H p. Recall that an operator T on a Hilbert space H is said to be a partial isometry, if T T is a projection on H. It is well-known that: T is a partial isometry, if and only if T T T T on H, if and only if T is a partial isometry on H, if and only if T T T T on H, if and only if T T is a projection on H. i.e., a partial isometry T is a unitary from T T phq onto T T phq. If T is a partial isometry on H, then the projection T T is called the initial projection of T, and the projection T T is called the final projection of T on H. Also, the (closed) subspaces T T phq and T T phq of H are called the initial subspace and the final subspace of T in H, respectively. If T is a partial isometry on H, then it is a unitary from its initial subspace onto its final subspace, in the sense that: T T 1 T T phq and T T 1 T T phq, where 1 means the identity operators on Hilbert (sub-)spaces (in H). Thus, if T has identical initial and final subspaces in H, then T T 1 T T, and hence, one can understand T as unitary in the operator subalgebra Bpq of BpHq. Notice that in Section 5 (and [2]), we observed that: α pf1 α pf2 α p f 1 f 2 for all f 1, f 2 P HpG p q, (6.1)

19 Free probability on Hecke algebras and certain group C -algebras α p f α p f for all f P HpG p q. (6.2) Theorem 6.1. Let f χ x be a generating element of HpG p q for x P G p, and a compact-open subgroup of G p. Assume x x in G p, and let α p f be the corresponding operator on the Hecke Hilbert space H p. α p f is a projection on H p ðñ µ p pq 1, and x P. (6.3) Proof. Recall that an operator T on an arbitrary Hilbert space H is a projection, if (i) T is self-adjoint in the sense that T T on H, where T is the adjoint of T, and (ii) T is idempotent in the sense that T 2 T on H. Observe now that α p f α p f α p α p pχ xq χ x α p f, by (6.2). Thus, the operator α p f is self-adjoint on H p. So, the given operator α p f satisfies the self-adjointness condition (i) automatically. Now observe that α p 2 f α p f f αp µ ppqχ on H x 2 p, (6.4) by (6.1), and by the assumption: x x in G p. So to satisfy the idempotence condition (ii), the operator α p f must satisfy α p µ ppq χ x 2 α p χ x on H p, (6.5) by (6.4). (ð) If µ p pq 1, and x P, then x, and hence, x 2, moreover, α p µ ppqχ x 2 α p χ α p χ x. Therefore, the relation (6.5) holds, and hence α p f is a projection on H p. (ñ) Suppose the relation (6.5) holds, and assume that either µ p pq 1, or x R in G p. Let x R in G p. Then, in general, x x 2, and hence, χ x 2 χ x. So, the relation (6.5) does not hold true, and it contradicts our assumption. Assume now that µ p pq 1. Then, clearly, µ p pqχ x 2 χ x, in general, thus the relation (6.5) does not hold either. It again contradicts our assumption. Therefore, we obtain the characterization α p f is an idempotent ðñ µ ppq 1, and x P. (6.6) By the self-adjointness of α p f, and by (6.5) and (6.6), one can conclude that: αp f is a projection on H p, if and only if µ p pq 1, and x P.

20 172 Ilwoo Cho The above characterization (6.3) shows that the generating elements f χ x of the normal-cored Hecke probability space phpg p q, ψ p q assign projections α p f on the Hecke Hilbert space H p, whenever f χ with µ p pq 1. (6.7) Let f j χ j be non-zero generating elements of phpg p q, ψ p q, where µ p p j q 1, equivalently, α p f j are projections on H p, by (6.3) and (6.7), for j 1, 2. Also, let f χ x P phpg p q, ψ p q, and α p f, the corresponding operator on H p, where x x in G p. Consider the following functional equation: Observe that Consider the equality (6.10) below: f f f 1 and f f f 2 on HpG p q. (6.8) f f µ p pqχ x 2 f f To satisfy (6.10), one must have that: in HpG p q. (6.9) µ p pqχ x 2 χ. (6.10) µ p pq 1, and x 2. (6.11) By (6.8), (6.9) and (6.10), we obtain the following theorem. Theorem 6.2. Let x 0 P G p, and 0,, compact-open subgroups of G p, where x x 0 in G p. If x 0 0 x 1 0 in G p, with µ p p 0 q 1 µ p pq, (6.12) then α p χ x0 0 is a partial isometry with its initial and final projections α p χ on H p. Proof. By (6.3) and (6.7), if µ p pq 1, then αχ p is a projection on H p. Assume now that x in G p, where µ p p 0 q 1, for some x 0 P G p. Then we have χ x 0 0 χ x0 0 χ x0 0 χ x0 0 µ p p 0 qχ x χ x χ on H p, by (6.9), (6.10) and (6.11). Similarly, one obtains that χ x0 0 χ x 0 0 χ x χ on H p. Thus, the operator αχ p x0 0 satisfies α p χx00 α p χx00 αχ p α p χx00 α p χx00 (6.13) on H p, by the assumption that x x 0 in G p. The relation (6.13) shows that the operator αχ p x0 0 is a partial isometry with its initial and final projections identified with the projection αχ p, on H p.

21 Free probability on Hecke algebras and certain group C -algebras The above necessary condition (6.12) shows that, whenever we fix a projection αχ p on H p (with µ p pq 1), one may take a partial isometry αχ p x0 0 on H p, whenever x 2 0 0, having its both initial and final projections α p χ. By the property of µ p, one automatically obtains that µ p px q µ p p 0 q µ p pq 1. Notice that the choice of 0, for a fixed, is not unique, i.e., one may have multi-partial isometries having both initial and final projections α p χ on mathfrakh p. Assume now that, for a fixed compact-open subgroup of G p with µ p pq 1, there are distinct compact-open subgroups j of G p such that x j j x 1 j and µ p p j q 1, (6.14) for some x j P G p, for j 1,..., N, for N P N. Then by (6.12), the operators αχ p xj are self-adjoint partial isometries having j their initial and final projections αχ p on H p, for j 1,..., N. And, by (6.14), one can understand the partial isometries αχ p xj as certain perturbed operators α p j χ x 1 j induced by x 1 j, satisfying (6.14) for all j 1,..., N, i.e., αχ p xj α p j χ on H p for all j 1,..., N. x 1 j The above equality holds by the quotient relation R p on the normal-cored Hecke Hilbert space H p. Let us denote these partial isometries αχ p xj α p j χ simply by T x 1 j for j j 1,..., N. Theorem 6.3. Let Tj be distinct partial isometries αχ p xj α p j χ satisfying x 1 j (6.14), whose initial and final projections αχ p, for j 1,..., N, for N P N, where j G p for j 1,..., N (and hence, G p, too, by (6.14)). Then the subgroup generated by ttj un (under the operator-multiplication on the operator algebra BpH p q) is group-isomorphic to a quotient group T N, T N F `ta j u N {ta 2 j e N u N where F `ta j u N is the free group generated by taj u N, and ta2 j e Nu N relator set of T N, where e N is the group-identity of T N. Proof. Let T j α p χ xj j be given as above, and let is the α p χ ph p q denote H p

22 174 Ilwoo Cho be the subspace of H p. Since αχ p is a well-defined projection on H p, its image H p is indeed a well-determined (closed) subspace of H p. Moreover, it is both the initial and final subspaces of Tj, by (6.12) and (6.14), for all j 1,..., N, in H p. So without loss of generality, one can understand Tj are operators in the operator (sub-)algebra BpH p q of BpH p q for j 1,..., N. By understanding ttj un as a subset of BpH p q, one can define the (multiplicative) subgroup T N (under operator multiplication on BpH p q), by the group generated finitely by ttj un, i.e., T N tt j u N D Ď BpH p q Ď BpH p q, (6.15) where xxy mean here the groups generated by sets X. Now let T N be the group, T N F `ta j u N {ta 2 j e N u N, (6.16) where FpXq mean the (noncommutative) free groups generated by sets X. Define now a morphism Ω : T N Ñ T N by the binary operation-preserving map such that Ω `T j aj for j 1,..., N (6.17) (with possible re-arrangements), where T N is in the sense of (6.15), and T N is in the sense of (6.16). Since both T N and T N have N-generators, the generator-and-operation-preserving morphism Ω of (6.17) is bijective. It also satisfies that 2 Ω `Tj a 2 j e N for all j 1,..., N. (6.18) Indeed, by definition, one has `T j 2 α p χ xj j 2 α p χxj j χ xj j α p µ pp jqχ x 2 j j α p χ 1 H p, where 1 H p means the identity operator on the subspace H p (in BpH p q) of H p. Thus, the formula (6.18) holds. Remark that even though 1,..., N are normal in G p, one has T i T j α p χ x1 1 χ x2 2 α p µ pp o 1,2 qχ x 1,2 1,2 α p µ pp o 2,1 qχ x 2,1 2,1 T j T i, in general, in T N, because x 1,2 x 2,1 in G p, while 1,2 2,1 in G p. Therefore, the bijective generator-and-operation-preserving morphism Ω also preserves the relations between T N and T N, and hence, it is a well-determined group-isomorphism from T N onto T N, i.e., two groups T N and T N are group-isomorphic.

23 Free probability on Hecke algebras and certain group C -algebras Notice that in the above theorem, the normality condition for 1,..., N is crucial. By the above theorem we obtain the following sub-structure theorem in α p phpg p qq in BpH p q. Theorem 6.4. Under the same hypothesis with the above theorem, the C -subalgebra generated by ttj un in BpH pq is -isomorphic to the group C -algebra C l 2 pt N q pt Nq in the sense of Section 2.3, i.e., C H p `T N -iso C l 2 pt N q pt Nq, (6.19) where C H pxq mean the C -subalgebras of BpHq generated by subsets X of BpHq over Hilbert spaces H. Proof. By the above theorem the (sub)group T N of (6.14) generated by tt j un (in BpH p q Ď BpH p q) is group-isomorphic to the group T N of (6.16), by the group-isomorphism Ω of (6.17), i.e., T N Group F `ta j u N {ta 2 j a j u N T N. Therefore, the group C -algebra `T C denote N C H p is -isomorphic to the group C -algebra `T N C T N of BpH p q C pt N q denote C l 2 pt N q pt N q C rupt N qs of B `l 2 pt N q, where u means the left-regular unitary representation in the sense of Section 2.3. Indeed, one can extend the group-isomorphism Ω of (6.17) under linearization, i.e., we have a morphism Ω o : C `T N Ñ C pt N q, such that Ω o nÿ t j T j def Nÿ t j Ω `T j nÿ t j u pa j q, for t j P C, j 1,..., n and n P N Y t8u (under C -topology). It is not difficult to check Ω o is a -isomorphism. The characterization (6.19) shows that α p phpg p qq contains group C -algebras ( -isomorphic to) C pt N q, for N P N, where T N are in the sense of (6.16), whenever there are compact-open normal subgroups with µ p pq 1, and distinct compact-open subgroups j with µ p p j q 1, satisfying x j j x 1 j for j 1,..., N.

24 176 Ilwoo Cho As in above theorems we assume is a normal compact-open subgroup of G p with µ p pq 1, and x j j x 1 j with µ p p j q 1 for all j 1,..., N. As a special case we consider the following conditions (6.20) and (6.21) below; suppose that the non-identity group elements x j of G p are self-invertible in the sense that: x j x 1 j ðñ x 2 j u p x 2 j, the group-identity of G p (6.20) for all j 1,..., N. And for the compact-open normal subgroup, take Then automatically we have that j x j for all j 1,..., N. (6.21) µ p p j q 1 for all j 1,..., N. Remark 6.5. Indeed, such group elements x j exist in G p. For instance, if we let ˆ a b x 1 a 2 P G p, b a for a, b P Q p, then x 2 u p in G p. So, one may take finitely many distinct elements x 1,..., x N in G p, for some N P N. Moreover, for a fixed normal subgroup of G p, we can take such x 1,..., x N in G p, which are not contained in. For instance, if is the normal core U Gp of U GL 2 pz p q, then we can take x 1 ˆ 2 3 and x ˆ 3 8 in G 1 3 p, satisfying x 1, x 2 R U Gp and hence, x 1 U Gp and x 2 U Gp are as in (6.21). Remark that ˆ 3 7 ˆ 2 7 x 1 x 2 x x 1, in G p. So, the group generated by tx 1 U G1, x 2 U G2 u is group-isomorphic to the noncommutative group Fpta 1, a uq{ta 1 2 j a j u 2. The corresponding operators Tj αχ p j are partial isometries on H p, whose initial and final projections are the projection αχ p on H p. Therefore, one can obtain the group, A E T N ttj αχ p un xj, (6.22)

25 Free probability on Hecke algebras and certain group C -algebras generated by ttj un, as a multiplicative subgroup of the operator algebra BpH p q, where H p αχ p ph p q is the subspace of H p. Note that T j T j α p χ i α p χ j α p α p µ ppxqχ x1 x 2 αp χ x1 x 2. (6.23) Assumption and Notation 6.6 (in short, AN 6.6 from below). In the rest of this paper if we write a group T N, then it means a group (6.22), which is a special case of the general construction (6.15), satisfying (6.23), i.e., j x j of (6.21), where x j satisfy (6.20), for j 1,..., N. But if we need to handle general cases as in (6.15) and (6.19), we will state clearly in the text. By the group-isomorphic relation in the general format of (6.15) with (6.16), a group T N of AN 6.6 is group-isomorphic to the group T N of (6.16), too. Recall that the group T N of (6.16) is defined to be the quotient group F `ta j u N {ta 1 j a j u N. In fact, the group T N is naturally group-isomorphic to the finitely presented group F N, C " * w F N tw j u N 2 N G j e N, and,, (6.24) w i w j w j w i i.e., T N Group F N. i, By the above discussions, we obtain the following refined results under AN 6.6. Corollary 6.7. Let T N be a group in the sense of (6.22) under AN 6.6. Then it is group-isomorphic to the finitely generated group F N of (6.24). Moreover, the group C -algebra C `T H p N is -isomorphic to the group C -algebra C l 2 pf N q pf N q, i.e., and T N Group F N def C H p C ta j u N, " aj a 1 j and a i a j a j a i `T N -iso C l 2 pf N q pf Nq. * N i, G, (6.25) Proof. By the discussion in the above paragraphs, the group T N of (6.16) is group-isomorphic to F N of (6.24), by (6.20), (6.21) and (6.23) (under AN 6.6). So one can define a morphism Ψ : T N Ñ F N by a generator-preserving bijection between the two finite sets, Ψpa j q w j for all j 1,..., N,

26 178 Ilwoo Cho such that Ψpa i a j q Ψpa i qψpa j q w i w j (under possible re-arrangements) for all i, j 1,..., N. Therefore, one has that T N Group T N Group F N. By the above group-isomorphic relations we obtain C H p `T N -iso C l 2 pt N q pt Nq -iso C l 2 pf N q pf N q. 7. FREE STRUCTURES ON `T C N In this section we study freeness conditions on our group C -algebras and their structure theorems. Now let be a fixed normal compact-open subgroup of G p, with µ p pq 1, and hence, the corresponding operator T αχ p is a projection on the Hecke Hilbert space H p, acting as the identity operator on the subspace H p T ph p q in H p. Assume further that there exist distinct self-invertible group elements x j P G p in the sense that: x 1 j x j, and distinct subsets j of G p with µ p p j q 1, such that j x 1 j x j for all j 1,..., N, as in AN 6.6. Then, by (6.12), the corresponding operators T j α p χ j are the partial isometries on H p with their initial and final projections identified with T αχ p, for j 1,..., N. We have seen in (6.19) and (6.25) the C -algebra `T C N is -isomorphic to the group C -algebra C pt N q generated by the finitely generated group, C G T N Group ta j u N, " a 2 j e N and a i a j a j a i * N i, Let s denote `T C N and C pt N q simply by C,N, and C N, respectively. Because of the -isomorphic relations between C,N and C N we sometimes use C,N and C N, alternatively, as a same object. However, whenever we emphasize such C -algebras C N are constructed from our Hecke representational setting we will precisely use the term C,N FREE-DISTRIBUTIONAL DATA ON C,N Let T N be the group in the general sense of (6.14) and C,N, the corresponding group C -algebra generated by T N (without AN 6.6). On the C -subalgebra C,N

27 Free probability on Hecke algebras and certain group C -algebras of BpH p q Ď BpH p q, define a linear functional, also denoted by ψ p, by a morphism satisfying ψ p `Tj ψp αχ p def xj ψ p `χxjj ϕp χ xj j:gp j ϕ p `χxjj χxjj pu p q µ p px j j X j q, µ p p j q (7.1) by the normality conditions for 1,..., N, where j:gp means the normal core Core Gp p j q of j in G p, as in Section 3 and where ψ p in the second equality def of (7.1) means the normal-cored linear functional ϕ p E p on the Hecke algebra HpG p q in the sense of (4.10) and ϕ p is the canonical linear functional on the normal Hecke algebra H Yp in the sense of (3.12). The pair C,N, ψ p becomes a well-determined a C -probability space in the sense of [12] and [13]. Definition 7.1. The C -probability space C,N, ψ p is called the (-concentrated- -C )-Hecke probability space on H p (or, on H p ). Remark that since one has that and hence, ψ p `T j µ p px j j X j q µ p p j q by (7.2), for all j 1,..., N. Notice here in (7.3) that x j j x 1 j for all j 1,..., N, j x 2 j for all j 1,..., N, (7.2) µ `x 1 p j X x 2 j µ p px 2 j q `x 1 µ p j X x 2 x P g X g 2 ô x gk 1 and x g 2 k 2, for some k 1, k 2 P j (7.3) and hence one has ô g 1 x k 1 and g 1 x gk 2 ô g 1 x P X g ô x P g p X gq, g X g 2 Ď g p X gq for g P G p. Similarly, x P g p X gq ô x gv with v k 1 gk 2, for some k 1, k 2 P ô x gk 1 and x g 2 k 2 ô x P g X g 2,

28 180 Ilwoo Cho and hence we have g p X gq Ď g X g 2 for g P G p. Therefore, g X g 2 g p X gq, for a compact-open subgroup of G p, and g P G p. So, the second equality of (7.3) indeed holds. It shows that the formula (7.3) can be re-written by ψ p `T j µp `x 1 j X x 2 j `x 1 µ p j ` X x 1 j µ p ` X x 1 j, i.e., ψ p `Tj µp ` X x 1 j (7.4) for all j 1,..., N, since µ p pq 1. So, one can conclude that ψ p `T j µp px j X q µ p px j X q µ p pq µ p px j X u p q `χxj `χxj ψ p ϕp, µ p pq (7.5) by the normality of, where u p is the group-identity of G, by the normality of in G p. By (7.5), it is not difficult to check that ψ p pt q ψ p `αp χ ψp pχ q χ pu p q µ pp X u p q µ p pq 1. It shows that the -Hecke probability space pc,n, ψ pq is unital in the sense that ψ p pt q ψ p 1 C 1,,N because T is the identity operator 1 C,N on H p in C,N.

29 Free probability on Hecke algebras and certain group C -algebras Observe now that nź ˆ ψ p Ti k ψ p k1 n χ x ik ik k1 ψ p ˆ n χ k1 x 1 i k by (7.2) ψ p µ p pq n 1 χ x 1 x 1...x 1 i 1 i 2 in by the normality condition for µ ppq n 1 µ p `pxin... x q 1 i1 X µ p pq µ p `pxin... x q 1 i1 X µ p pq µ p `pxin... x i2 x q 1 i1 X (7.6) refining (7.4) and (7.5). The above formulas (7.5) and (7.6) are also obtained under AN 6.6, too. Theorem 7.2. If pi 1,..., i n q P t1,..., Nu n, for n P N, then ψ p nź k1 ź Ti k µ p n 1 x in k 1 X. (7.7) k0 Proof. The proof of (7.7) is done by formula (7.6). So one obtains the following corollary immediately. Corollary 7.3. Under AN 6.6, if pi 1,..., i n q P t1,..., Nu n for n P N, then nź ˆˆn 1 ψ p Ti k.µ p Π x i n k X. (7.8) k0 k1 The above formula (7.7) (or (7.8)) characterizes the free-distributional data of our partial isometries ttj un (resp., under AN 6.6). For pi 1,..., i n q P t1,..., Nu n, for n P N, consider now the free cumulants, ÿ ˆ ˆ k n `Ti1, Ti 2,..., Ti n ˆψ p Π o T i jpv j µ `0 V, 1 V πpncpnq Π V Pπ by the Möbius inversion of Section 2.2, where k n p...q means the free cumulant for ψ p on C 2,N