Some remarks on traces on the infinite-dimensional Iwahori--Hecke algebra
aa r X i v : . [ m a t h . R T ] J a n Some remarks on traces on theinfinite-dimensional Iwahori–Hecke algebra
Yury A. Neretin The infinite-dimensional Iwahori–Hecke algebras H ∞ ( q ) are direct limits of theusual finite-dimensional Iwahori–Hecke algebras. They arise in a natural way as con-volution algebras of bi-invariant functions on groups GLB( F q ) of infinite-dimensionalmatrices over finite-fields having only finite number of non-zero matrix elements underthe diagonal. In 1988 Vershik and Kerov classified all indecomposable positive traceson H ∞ ( q ). Any such trace generates a representation of the double H ∞ ( q ) ⊗ H ∞ ( q )and of the double GLB( F q ) × GLB( F q ). We present constructions of such representa-tions; the traces are some distinguished matrix elements. We also obtain some (simple)general statements on relations between unitary representations of groups and repre-sentations of convolution algebras of measures bi-invariant with respect to compactsubgroups. Let G be a topological group.Denote by M ( G ) the algebra of finite compactly supported complex-valuedmeasures on G . The addition in M ( G ) is the addition of measures, the mul-tiplication is the convolution. Namely, let µ , µ be measures supported bycompact sets L , L respectively. Consider the measure µ × µ on L × L . The convolution µ ∗ µ is the pushforward of µ × µ under the map ( g , g ) g g from L × L to G . We also define an involution µ µ ⋆ in M ( G ). Namely, µ ⋆ is the pushforward of the the complex conjugate measure µ under the map g g − , ( µ ∗ µ ) ⋆ = µ ⋆ ∗ µ ⋆ . We also define a transposition that send µ to its image under the map g g − . We have ( µ ∗ µ ) t = µ t ∗ µ t , the involution is anti-linear map and the transposition is linear. Remark.
If a group G is finite, then M ( G ) is the group algebra of G . ⊠ Let ρ be a unitary representation of G in a Hilbert space H ( Hilbert spacesassumed to be separable ). For any measure µ ∈ M ( G ) denote by ρ ( µ ) theoperator ρ ( µ ) = Z G ρ ( g ) dµ ( g ) . This determines a ∗ -representation of the algebra M ( G ), ρ ( µ ) ρ ( µ ) = ρ ( µ ∗ µ ) , ρ ( µ ) + ρ ( ν ) = ρ ( µ + ν ) , ρ ( µ ⋆ ) = ρ ( µ ) ∗ . Supported by the grant of FWF (Austrian science fund) P31591. K ⊂ G be a compact subgroup. By δ K we denote the probabilisticHaar measure on K regarded as an element of M ( G ). Denote by M ( G//K ) thesubalgebra of M ( G ) consisting of measures, which are invariant with respectto left and right shifts by elements of K . Clearly, for any µ ∈ M ( G ) we have δ K ∗ µ ∗ δ K ∈ M ( G//K ) and for an element ν ∈ M ( G ) the following identityholds: ν ∈ M ( G//K ) ⇐⇒ δ K ∗ ν ∗ δ K = ν. Clearly, the subalgebra M ( G//K ) is closed with respect to the involution.Let ρ be a unitary representation of G in a Hilbert space H . Denote by H K ⊂ H the subspace consisting of all K -fixed vectors, denote by P K theoperator of orthogonal projection to H K . Clearly, P K := ρ ( δ K ) . Therefore, for any ν ∈ M ( G ) we have P K ρ ( ν ) P K = ρ ( ν ) . Therefore, such operators have the following block form with respect to thedecomposition H = H K ⊕ ( H K ) ⊥ : ρ ( ν ) = (cid:18)e ρ ( ν ) 00 0 (cid:19) . (1.1)So for any unitary representation ρ of G we get a ∗ -representation e ρ ( · ) of thealgebra M ( G//K ) in the Hilbert space H K .Denote by URep( G ) K the set of equivalence classes of unitary representa-tions of the group G such that H K is a cyclic subspace . Denote by Rep( G//K )the set of ∗ -representations of the algebra M ( G//K ). The following statementis obvious (see below Proposition 2.1)
The map R es G//K : ρ e ρ from URep( G ) K to Rep( M ( G//K )) is injective. An inverse construction e ρ ρ (if e ρ is contained in the image of R G//K ) issemi-explicit: having a representation of M ( G//K ) one can define a reproduc-ing kernel space and a representation of G in this space (see Proposition 2.2).Generally speaking, a positive-definiteness of a kernel is a non-trivial questionand a description of the image of R es G//K also is non-trivial . For our purposesthe following statement is sufficient: If G is a compact group or a direct limit of compact groups, then the map R es G//K is a bijection , see Propositions 2.3–2.5.Recall also two variations of these definitions. Let G be a unimodular locallycompact group , let λ be a Haar measure on G . Let K ⊂ G be a compact I.e., linear combinations of vectors ρ ( g ) ξ , where ξ ranges in H K and g ranges in G , aredense in H . For instance, if G is a semisimple Lie group and K is its maximal compact subgroup. A locally compact group is called unimodular if its Haar measure is two-side invariant,see, e.g., [17], Subsect. 9.1. C ( G ) the convolution algebra of all compactly supportedcontinuous functions on G . This algebra is a subalgebra in M ( G ), namely, forany function ϕ ∈ C ( G ) we assign the measure ϕ ( g ) µ ( g ). By C ( G//K ) ⊂ C ( G )we denote the subalgebra of K -bi-invariant functions, ϕ ( k gk ) = ϕ ( g ) for k , k ∈ K .Next, let G be a unimodular totally disconnected locally compact group, k be an open compact subgroup. Denote by A ( G//K ) the convolution algebraconsisting of K -bi-invariant compactly supported locally constant functions.Clearly, in this case A ( G//K ) = C ( G//K ) = M ( G//K ) . Convolution algebras M ( G//K ) are a usual tool of representation theory,see, e.g., [19], [9], [1], [12], [13], [5], [2], [18], [20], Chapter 5.
Consider a finite field F q with q elements.Let G be the group GL( n, F q ) of invertible matrices of order n over F q . Let K = B ( n ) be the subgroup of upper triangular matrices. The algebra M ( G//K )was described by Iwahori [12]. Denote by s m ∈ GL( n, F q ) the operator thattransposes basis vectors e m and e m +1 in F nq and fixes other e j . So m = 1, 2,. . . , n −
1. Set σ m := δ K ∗ s m ∗ δ K . Then the elements σ m generate the algebra H n ( q ) := M (cid:0) GL( n, F q ) //B ( n ) (cid:1) and relations are σ m = ( q − σ k + q or ( σ m + 1)( σ m − q ) = 0; (1.2) σ m σ m +1 σ m = σ m +1 σ m σ m +1 (1.3) σ m σ l = σ l σ m if | m − l | >
1. (1.4)The dimension of H q = n !. The involution and the transposition are determinedby σ ⋆m = σ m , σ tm = σ m . By the construction q is a power of a prime. However, the algebra with rela-tions (1.2)–(1.4) makes sense for any q ∈ C , for q = 1 we get the group algebraof the symmetric group S n . For all q that are not roots of units algebras H n ( q )are isomorphic. Therefore they have the same dimensions of representations. Following Vershik and Kerov [35], denote by GLB( F q ) thegroup of all infinite invertible matrices over F q having only finite number ofnonzero matrix elements under the diagonal .Denote by B ( ∞ ) the subgroup of GLB( F q ) consisting of upper triangularmatrices. For a matrix g ∈ K its diagonal elements g jj are contained in F × q := F q \ g ij , where i < j are contained in F q . So the set B ( ∞ ) is a There are several approaches to ’representation theory of infinite-dimensional groups overfinite fields’, see a discussion of different works in [27], Subsect. 1.11. F × q and a countable number of copiesof F q . We equip K with the product topology and get a structure of a compacttopological group. We take uniform probabilistic measures on the sets F × q , F q and equip the group B ( ∞ ) with the product measure. It is easy to verify thatwe get the Haar measure on B ( ∞ ).The space GLB( F q ) is a disjoint union of a countable number of cosets gB ( ∞ ). We equip GLB( F q ) with the topology of disjoint union, this determinesa structure of a unimodular locally compact topological group on GLB( F q ).Equivalently, a sequence g ( α ) ∈ GLB( F ) converges to g if the following twoconditions hold:— for each i , j we have g αij = g ij for sufficiently large α ;— there exists n such that for all i , j such that i < j and j > n we have g αij = 0 for all α .Denote by GLB( n, F q ) ⊂ GLB( F q ) the subgroup consisting of matrices g satisfying the condition: g ij = 0 for all pairs ( i, j ) such that j > n , i < j .In other words this group is generated by GL( n, F q ) and B ( ∞ ). We get anincreasing family of compact subgroups in GLB( F q ), and GLB( F q ) is the directlimit GLB( F q ) = lim n −→∞ GLB( n, F q ) . The Iwahori–Hecke algebra H ∞ ( q ) := M (cid:0) GLB( F q ) //B ( ∞ q ) (cid:1) = A (cid:0) GLB( F q ) //B ( ∞ q ) (cid:1) is the algebra with generators σ , σ , σ , . . . and the same relations (1.2)–(1.4).Also, it is the direct limit H ∞ ( q ) = lim n −→∞ H n ( q ) . Again, algebras H ∞ ( q ) are well defined for all q ∈ C . By default we assume q > . H ∞ ( q ). A trace χ on an associative algebra A withinvolution is a linear functional A → C such thata) χ ( AB ) = χ ( BA ) for any A , B ∈ A ;b) χ ( A ⋆ ) = χ ( A ).c) χ ( A ⋆ A ) > A ∈ A ( q ).The set of all traces is a convex cone: if χ , χ are traces, then for any a , a > a χ + a χ is a trace. A trace χ is normalized if χ (1) = 1.A trace ν is indecomposable if it does not admit a representation of the form ν = a χ + a χ , where χ , χ are traces non-proportional to ν and a , a > all indecomposable traceson the infinite-dimensional Iwahori–Hecke algebras H ∞ ( q ). See also continua-tions of this work in [21], [4], [10]. Kerov died in 2020, the work was not completely published, the text [36] was based onhis posthumos notes. λ ν we denote ζ λ,µ := σ µ − σ µ − . . . σ λ . Consider a partition ν of n , n = ν + ν + . . . , ν > ν > · · · > . We set λ j := ν + · · · + ν j , and consider the following elements of the Iwahori–Hecke algebra ζ λ := . . . ζ λ ,λ ζ λ ,λ ζ λ ,λ (all factors commute). For the following two statements, see [30], [8], Sect.8.2.— The elements ζ λ form a basis of the space H n ( q ) / [ H n ( q ) , H n ( q )] , where [ H n ( q ) , H n ( q )] ⊂ H n ( q ) is the subspace generated by all commutators ab − ba , where a , b range in H n ( q ). Therefore,— a trace on H n ( q ) is determined by its values on elements ζ λ . This implies that the same statement is valid for H ∞ ( q ). Next,— By [34], for any indecomposable trace χ on H ∞ ( q ) we have χ ( ζ λ ) = Y j ζ λ j , (1.5) and for any m > χ ( ζ [ λ + m,λ + m ] ) = χ ( ζ [ λ ,λ ] ) . (1.6)Therefore, any trace χ on H ∞ ( q ) is determined by its values on elements ζ m := ζ [1 ,m ] . Notice, that ζ = 1. Theorem 1.1 (Vershik, Kerov [34])
Let q > . Indecomposable traces χ α,β,γ on H ∞ ( q ) are enumerated by collections of parameters α > α > . . . > , β > β > . . . > , γ > X α i + X β j + γ = 1 . The value of χ α,β,γ on ζ m is given by the formula χ α,β,γ ( ζ m ) = 1 q − X µ > , µ > , . . . P kµ k = m Y k > ( q k − µ k k µ k µ k ! Y k > p k ( α, β ) µ k , (1.7)5 here p k ( α, β ) := X i α ki + ( − k +1 X i β ki (1.8) denote the super-Newton sums . Remark.
Recall that symmetric functions [20] can be represented as poly-nomials of the Newton sums p k ( α ) := P α ki . Supersymmetric functions arepolynomials in super-Newton sums p k ( α, β ), see [16], [32], [3], Chapter 2. Remark.
For q = 1 (i.e., for the infinite symmetric group) the expression(1.7) has a removable singularity, χ α,β,γ ( ζ m ) = p m ( α, β ) for m > . This special case of Theorem 1.1 is the Thoma theorem [33] (see, also [3], Chap-ter 4): irreducible normalized characters of the infinite symmetric group havethe following form χ α,β,γ ( g ) = Y m > p m ( α, β ) r m ( g ) , where r m ( g ) is the number of cycles of g of order m . ⊠ H ∞ ( q ) ⊗ H ∞ ( q ). Let A be a ∗ -algebra with unit, let χ be a trace on A . Denote by A ◦ the algebraanti-isomorphic to A , it coincides with A as a linear space, multiplication isgiven by A ⋄ B := BA . The following formula determines an inner product on A : h A, B i χ = χ ( B ⋆ A ) . Denote by A χ the corresponding Hilbert space. Assume that operators of leftmultiplication are bounded in A χ . Then we have an action of A on A by leftmultiplications and the action of A ◦ by right multiplications. These actionscommute, so we get a representation of the tensor product A ⊗ A ◦ , τ χ ( A ⊗ B ) X := AXB, where X ∈ A , A ∈ A , B ∈ A ◦ . Remark.
Representations τ χ of A ⊗ A ◦ are not arbitrary representations,they satisfy the condition: τ χ ( A ⊗
1) 1 = τ χ (1 ⊗ A ) 1 . ⊠ Notice, that χ ( A ) = (cid:10) τ χ ( A ⊗
1) 1 , (cid:11) χ . (1.9)So a trace on A is a certain matrix element of a certain ( A , A ◦ )-bimodule.Let us return to a discussion the Iwahori-Hecke algebras. Notice that atransposition in H ∞ ( q ) is an anti-isomorphism, so we can regard H ∞ ( q ) as a( H ∞ ( q ) ⊗ H ∞ ( q ))-module. So have a structure of a Hilbert algebra in the sense of Dixmier (see [6], § I.6, [7], A.54). emma 1.2 We have an isomorphism of algebras M (cid:16) GLB( F q ) × GLB( F q ) //B ( ∞ ) × B ( ∞ ) (cid:17) ≃≃ M (cid:16) GLB( F q ) //B ( ∞ ) (cid:17) ⊗ M (cid:16) GLB( F q ) //B ( ∞ ) (cid:17) = H ∞ ( q ) ⊗ H ∞ ( q ) . More generally, for any locally compact group G with a countable base oftopology and an open compact subgroup K , the following algebras are isomorphic M (cid:0) G × G//K × K (cid:1) ≃ M ( G//K ) ⊗ M ( G//K ) . Lemma 1.3
For any indecomposable trace χ α,β,γ on H ∞ ( q ) the operators ofleft-right multiplication τ χ ( A ⊗ B ) X := AXB are bounded in H ∞ ( q ) χ . The first statement is obvious, the second statement is clear from the condi-tion (1.2). Indeed, the spectrum of a self-adjoint operator of multiplication bya generator σ m consists of points 1 and q , so its norm is max(1 , q ).Our Proposition 2.5 implies the following corollary: Corollary 1.4 ( see, [35]) Let q = p l , where p is a prime. Then for any inde-composable trace χ α,β,γ there exists an irreducible representation ρ of the double GLB( F q ) × GLB( F q ) , for which R es( ρ ) is isomorphic to the representation τ χ of H ∞ ( q ) ⊗ H ∞ ( q ) . Remark.
To avoid a misleading, we must say some remarks. For a type Itopological group G any irreducible representation of G × G is a tensor product π ⊗ π of irreducible representations of two copies of G (see, e.g., [7], 13.1.8).The group GLB( F q ) is not of type I and a similar implication is false. Accord-ing [34], for an indecomposable trace χ , the operator algebra generated by therepresentation of H ∞ ( q ) ⊗ V = H ∞ ( q ) is a finite Murray–von Neumannfactor. The representation of 1 ⊗ H ∞ ( q ) also generates a factor, which is thecommutant of the first factor. The representation of the double H ∞ ( q ) ⊗ H ∞ ( q )is irreducible. The trace of the unit operator is finite (= 1), so it is a factor ofa type I n or II . But H ∞ ( q ) has only two irreducible finite dimensional rep-resentations, both are one-dimensional . The first is the trivial representation( α = 1, other α ’s and β ’s are zero), the second is the representation sending all σ j to − β = 1, other β ’s and α ’s are 0). In the remaining cases we have II -factors. The representations of GLB( F q ) corresponding to the one-dimensionalcharacters are the trivial representation and the Steinberg representation (see, Indeed, the algebra H n ( q ) is isomorphic to the group algebra of the symmetric group S n , so dimensions of their irreducible representations coincide. So they have two one-dimensional representations, all other representations have dimensions > n −
1. Thereforefinite-dimensional irreducible representations of H ∞ ( q ) are one-dimensional. χ α,β,γ correspond to representations ofGLB( F q ) generating II ∞ -factors. ⊠ Our next purpose is to construct explicitly representations τ χ of H ∞ ( q ) ⊗H ∞ ( q ). We consider only the case γ = 0 . R -matrix. Let V be a Hilbert space equipped with an orthonormalbasis v j , where j ranges in non-zero integers. Consider R -matrix given by theformula R := − X i< e ii ⊗ f ii + q X i> e ii ⊗ f ii −− √ q X i = j,i =0 ,j =0 e ji ⊗ f ij + ( q − X i
1) in the middle factor ( V ⊗ V ) ⊗ ( W ⊗ W );— 1 ⊗ ( ∞− j − in the last factor.We get a representation of H ∞ ( q ) in X , denote it by A A ( l ) . In the same way we define operators R right j ( j +1) acting by twisted permutations offactors W and a representation of the second copy of the algebra H ∞ ( q ). Theorem 1.5 a) For any A ∈ H ∞ ( q ) , h A ( l ) Ξ , Ξ i X = χ α,β, ( A ) . b) Moreover, the representation of H ∞ ( q ) ⊗ H ∞ ( q ) in the cyclic span of Ξ is equivalent to the representation in H ∞ ( q ) χ . In Section 2 we show thata unitary representation ρ ∈ URep( G ) K is uniquely determined by the corre-sponding representation e ρ of the algebra M ( G//K ) (Proposition 2.1) and de-scribe a quasi-explicit way of realization of ρ (Proposition 2.2) in a reproducingkernel space constructed by e ρ . Next, we show that for a direct limit L ofcompact groups any ∗ -representation of M ( L//K ) corresponds to a unitary rep-resentation of L (Proposition 2.5). All these statements are obvious or verysimple, however I could not find a source for formal references.In Section 3 we prove Theorem 1.5, this proof does not depend on Section2. On reproducing kernel spaces, see, e.g., [23], Sect. 7.1 and Subsect. 7.5.15. .9. Some comments. The group GLB( F q ) is a locally compact group,whose properties are partially similar to properties of ’infinite dimensional’groups. The construction of Subsect. 1.7 looks like a relatively usual con-struction related to infinite dimensional groups. However, a behavior of doublecoset algebras is unusual.As an example of the usual behavior, consider G α ( n ) := GL( α + n, R ), K α ( n ) := O( n ). For α = 0 the algebra C (cid:0) G ( n ) //K ( n ) (cid:1) is commutative andaccording Gelfand [9] this implies sphericity of the subgroup O( n ) in the groupGL( n, R ). However, even the algebra C (cid:0) SL(2 , R ) // SO(2) (cid:1) is a nontrivial ob-ject, a multiplication is defined in terms of a hypergeometric kernel, see, e.g.,[18]. For α > C (cid:0) G α ( n ) //K α ( n ) (cid:1) at the present time seem completelyincomprehensible.We have a natural map of double coset spaces K α ( n ) \ G α ( n ) /K α ( n ) → K α ( n + 1) \ G α ( n + 1) /K α ( n + 1) . but K α ( n + 1) is strictly larger than K α ( n ) and this map does not induce ahomomorphism from C (cid:0) G α ( n ) //K α ( n ) (cid:1) to C (cid:0) G α ( n + 1) //K α ( n + 1) (cid:1) . However,a limit object exists (see [28], [24]) and it has a structure of a semigroup. Namely,there is a natural multiplication on the space of double cosetsΓ α ( ∞ ) = K α ( ∞ ) \ G α ( ∞ ) /K α ( ∞ ) , it admits a reasonable description (see [22], Section IX.4), and the semigroupΓ α ( ∞ ) acts in subspaces of K α ( ∞ )-fixed vectors of unitary representations of G α ( ∞ ) = GL( ∞ , R ). Moreover, this situation is typical for infinite dimensionalgroups and allows to produce unconventional algebraic structures (see, e.g.,[26]).In the context of the present paper, we have increasing groups G ( n ) and con-stant compact subgroup K , for this reason we have embeddings M ( G ( n ) //K ) →M ( G ( n + 1) //K ) and a direct limit M ( G ( ∞ ) //K ). Moreover, we have compre-hensive prelimit algebras M ( G ( n ) //K ). This situation is unusual among infinite-dimensional groups whose representation theories were topic of considerations ofmathematicians. Certainly, additionally there is a group GL of two-side-infinitealmost triangular matrices over F q and its symplectic, orthogonal, and ’unitary’subgroups (in all cases the subgroup K consists of upper triangular matrices),the author does not see a way to extend this list. G . To avoid an appearanceof exotic measures let us fix a class of topological groups under considerations.Recall that a topological space is Polish if it is homeomorphic to a completemetric space. A topological group is
Polish if it is a Polish topological space.10onsider a sequence G ⊂ G ⊂ . . . of Polish groups. We say that the group G := ∪ G j =: lim −→ G j is their direct limit . We equip G with the topology of a direct limit, a set U ⊂ G is open if all intersections U ∩ G j are open in G j . Generally speaking suchtopologies are not metrizable. If a subset K ⊂ G is compact, then K ⊂ G j forsome j . Remark.
A Polish group G is a direct limit of Polish groups, G j = G . ⊠ Let G = lim G j be a direct limit of Polish groups. Denote by M ( G ) thealgebra of all compactly supported Borel complex-valued measures on G , thisalgebra is a direct limit of algebras M ( G ) = lim −→ M ( G j ) , Notice that such measures are objects of classical measure theory on compactmetrizable spaces (see, e.g, [31], Chapter 4). We say that a sequence µ j ∈ M ( G )converges to µ if there is a compact subset L ⊂ G such that µ j ( G \ L ) = 0 forall j and µ j weakly converge to µ on L .Let K ⊂ G be a compact subgroup, without loss of generality we can assume K ⊂ G . Denote by M ( G//K ) ⊂ M ( G ) the subalgebra of K -bi-invariantcompactly supported measures. Proposition 2.1
Let G be a direct limit of Polish groups. If a ∗ -representation τ of the algebra M ( G//K ) in a Hilbert space V can be represented as R es G//K ( ρ ) and R es G//K ( ρ ′ ) , then ρ ≃ ρ ′ . We formulate a stronger version of the statement including a way of a recon-struction of ρ . Consider the homogeneous space K \ G , denote by z the basepoint of this space, i.e., the coset K ·
1. Define a kernel L ( z, u ) on K \ G × K \ G taking values in bounded operators V → V by L ( x, y ) = τ (cid:0) δ K ∗ hg − ∗ δ K (cid:1) , where z g = x , z h = y (the result does not depend on a choice of g , h ). Notice that the kernel is G -invariant, L ( xr, yr ) = L ( x, y ) , for r ∈ G .Consider the space ∆( K \ G, τ ) of finitely supported functions K \ G → V .For a vector v ∈ V denote by vδ z ( x ) the function, which equals v at the point z and 0 at over points. So, our space ∆( K \ G ) consists of finite linear combinations n X j =1 v j δ z j ( x ) . G acts in the space ∆( K \ G, τ ) by shifts of the argument. Definea sesquilinear linear form on ∆( K \ G, τ ) setting DX v i δ a i , X w j δ b j E = X i,j (cid:10) L ( a i , b j ) v i , w j (cid:11) V . (2.1)If for all vectors P v i δ a i we have DX v i δ a i , X v i δ a i E = X i,j (cid:10) L ( a i , a j ) v i , v j (cid:11) V > , (2.2)then h· , ·i is an inner product, we get a structure of a pre-Hilbert space and takethe corresponding Hilbert space ∆( K \ G, τ ). Since the kernel is G -invariant,shifts on K \ G induce unitary operators in ∆( K \ G, τ ). Proposition 2.2
Let G be a direct limit of Polish groups. Let τ = R es G//K ( ρ ) .Then (2.2) holds and the representation of G in ∆( K \ G, τ ) is equivalent to ρ . Proof.
Consider a unitary representation ρ ∈ URep( G ) K in a Hilbertspace H and the corresponding representation τ of M ( G//K ) in V := H K . Weconsider the map ∆( G//K, τ ) → H determined by J : X i v i δ z g i ρ ( g − i ) v i . (2.3)We have (cid:10) ρ ( g − ) v, ρ ( h − ) w (cid:11) H = (cid:10) ρ ( g − ) ρ ( δ K ) v, ρ ( h − ) ρ ( δ K ) w (cid:11) H == (cid:10) ρ ( δ K ) ρ ( h ) ρ ( g − ) ρ ( δ K ) v, w (cid:11) H = (cid:10) ρ ( δ K ∗ hg − ∗ δ K ) v, w (cid:11) H == (cid:10) ρ ( δ K ∗ hg − ∗ δ K ) v, w (cid:11) H K . Therefore D J (cid:16)X i v i δ z g i (cid:17) , J (cid:16) w j δ z g j (cid:17)E H = DX i ρ ( g − i ) v i , X j ρ ( h − j ) w j E H == X i,j (cid:10) ρ ( δ K ∗ h j g − i ∗ δ K ) v i , w j (cid:11) H K == X i,j (cid:10) L ( z g i , z h j ) v i , w j (cid:11) H K . So the map J induces an inner product on the space ∆( K \ G, τ ), and thisinner product coincides with the sesquilinear form (2.1). So (2.2) is positive andthe map (2.3) determines a unitary operator ∆( K \ G, τ ) → H . (cid:3) Remark. If τ is an arbitrary ∗ -representation of M ( G//K ), then we canrepeat the definition of the space ∆( K \ G, τ ), but positivity (2.2) of an operator-valued reproducing kernel L ( x, y ) can be a heavy problem. ⊠ roposition 2.3 For a compact group L with a countable base any ∗ -represen-tation τ of the algebra C ( L//K ) has the form R es L//K ( ρ ) . Corollary 2.4
The same statement holds for the algebra M ( L//K ) . Proof.
Consider the set b L of all irreducible representations ρ α of the group L defined up to equivalence , denote by H α (finite-dimensional) spaces of theserepresentations, by Mat( H α ) the algebras of all operators in H α .Consider the Fourier transform on L , see, e.g., [17], Subsect. 12.2. Namely,for f ∈ C ( L ) the operator-valued function F f ( α ) := ρ α ( f ) ∈ Mat( H α ) , where α ∈ b L, is called the Fourier transform F f of f . By the definition, the Fourier transformsends convolutions to point-wise products. The F is injective, a convenientdescriptions of the Fourier-image of C ( G ) and of the induced convergence in theFourier-image are unknown. In any case, functions α
7→ k ρ α ( f ) k are bounded;the image of the Fourier transform contains the set of all finitary functions (i.e.,functions whose elements are zeros for all but a finite number of α ). Moreover,finitary functions are dense in the Fourier-image. The uniform convergence offunctions f j → f implies convergences ρ α ( f j ) → ρ α ( f ).Denote by χ β the character of ρ β . Then F χ β ( α ) = ρ α ( χ β ) = ( (dim H β ) − · , if α = β ;0 otherwise . Next, consider the subset b L K ⊂ b L consisting of representations having non-zero K -fixed vectors. Define the Fourier transform F K on C ( L//K ) as operator-valued function F K f ( α ) := e ρ α ( f ) ∈ Mat( H Kα ) . These operators are left upper blocks of matrices (1.1). Properties of F can eas-ily translated to the corresponding properties of F K . In particular, the Fourier-image F K ( C ( L//K )) contains a copy of each algebra Mat( H Kα ), it consist offunctions supported by one point α . Notice that elements ζ β := dim( H β ) · χ β ∗ δ K ∈ C ( G )are commuting idempotents, ζ β ∗ ζ γ = ( ζ β , if β = γ ;0 otherwise . Their Fourier-images are F K ζ β ( α ) = ( , if α = β ;0 otherwise . This set is finite if the group L is finite, otherwise it is countable. ∗ -representation τ of the algebra C ( G//K ) ≃ F K ( C ( G//K ))in a Hilbert space V . Then operators τ ( ζ α ) are commuting projectors, denote by V α their images, these subspaces are pairwise orthogonal. Moreover, V = ⊕ α V α .Indeed, let v ∈ ( ⊕ α V α ) ⊥ . Then it is annihilated by all subalgebras Mat( H Kα )in F K ( C ( G//K )). But their sum is dense in the Fourier-image, therefore it isannihilated by the whole algebra F K ( C ( G//K )). Thus, v = 0.Thus, it is sufficient to construct a desired extension for each summand V α .So, without loss of generality we can assume V = V α . Then for all β = α operators τ ( ζ β ) are zero and τ is zero on each Mat( H Kβ ). So C ( L//K ) (cid:14) ker τ ≃ Mat( H Kα ) . Any representation of a matrix algebra is a direct sum of irreducible tautologicalrepresentations. By definition, each summand arises from the representation ρ α of L . (cid:3) Let L be a direct limit L := lim −→ L j of compact groups, let all L j have countable bases of topology. Then any ∗ -representation τ of the algebra M ( L//K ) has the form R es L//K ( ρ ) . Proof.
It is sufficient to check the positivity of (2.2). Fix an expression P v i δ a i . Since the summation is finite, we have only finite collection { a j } ⊂ K \ L . Therefore the subset is contained in some K \ L j . But the prelimit group L j is compact, and we can apply Proposition 2.3. (cid:3) H ∞ ( q ) Consider the Hilbertspace V as in Subsect. 1.7 equipped with the basis v j , the and the copy W of V . Consider the tensor product V ⊗ W and a unit vector ξ := X j =0 a / j v j ⊗ w j , where a j > X a j = 1.Consider the tensor power( V ⊗ W ) ⊗ n ≃ V ⊗ n ⊗ W ⊗ n and the vector Ξ := ξ ⊗ n ∈ ( V ⊗ W ) n . Denote elements of the natural orthonor-mal basis in our space by η (cid:20) i . . . i n j . . . j n (cid:21) := ( v i ⊗ v j ) ⊗ · · · ⊗ ( v i n ⊗ v j n ) , in this notation Ξ = X i ,...,i n ∈ Z \ n Y k =1 a / i k · η (cid:20) i . . . i n i . . . i n (cid:21) .
14e say that an operator in ( V ⊗ W ) ⊗ n is V -diagonal if it has the form D V (Φ) η (cid:20) i . . . i n j . . . j n (cid:21) = Φ( i , . . . , i n ) η (cid:20) i . . . i n j . . . j n (cid:21) , where Φ( i , . . . , i n ) is a bounded function ( Z \ n → C . For σ ∈ S n denote by T V ( σ ) the permutation of factors V in ( V ⊕ W ) n corresponding to σ , T ( σ ) η (cid:20) i . . . i n j . . . j n (cid:21) = η (cid:20) i σ − (1) . . . i σ − ( n ) j . . . j n (cid:21) . Denote by A V the algebra of operators in ( V ⊗ W ) ⊗ n generated by per-mutations T V ( σ ) and operators D V (Φ). Any element of this algebra can berepresented as a linear combination of the form X σ ∈ S n T V ( σ ) D Vσ (Φ) or X σ ∈ S n D Vσ ( e Φ) T V ( σ ) . In the same way we define an algebra A W , it consists of similar operators actingon the factors W . These two algebras commute. The map X σ ∈ S n T V ( σ ) D Vσ X σ ∈ S n T W ( σ ) D Wσ . We define a linear anti-automorphism (’transposition’) in A V by (cid:16) X σ ∈ S n T V ( σ ) D Vσ (Φ) (cid:17) t = X σ ∈ S n D Vσ (Φ) T V ( σ − ) . Proposition 3.1 a) For any elements A V , B V ∈ A V we have (cid:10) A V B V Ξ , Ξ (cid:11) = h B V A V Ξ , Ξ (cid:11) . (3.1)b) For any element B W ∈ A W we have B W Ξ = ( B V ) t Ξ . (3.2)c) For any elements A V ∈ A V , B W ∈ A W , (cid:10) A V B W Ξ , Ξ (cid:11) = (cid:10) ( B V ) t A V Ξ , Ξ (cid:11) . (3.3)So we come to the case of Hilbert algebras discussed in Subsect. 1.5. Thefunction χ ( A V ) = h A V Ξ , Ξ i is a trace on the algebra A V (an explicit formula is (3.4)). The A V -cyclic spanof Ξ is identified with A V χ , the map A V → A V χ is A V → A V Ξ. The rightaction of our algebra on A V χ is given by X X ( A W ) t .15he construction of Subsect. 1.7 gives an embedding H ∞ ( q ) → A V and an action of H ∞ ( q ) ⊗ H ∞ ( q ) on the A V χ . In Subsect. 3.3 we will showthat this embedding induces the Vershik–Kerov trace.
We use notation I := ( i , . . . , i n ), J :=( j , . . . , j n ). For σ ∈ S n denote by Ω( σ ) the set of all I invariant with respectto σ . In other words we decompose σ into a product of independent cycles, andmap k i k is constant on cycles. Lemma 3.2 (cid:10) T V ( σ ) D V (Φ) Ξ , Ξ (cid:11) = X I =( i ,...,i n ) ∈ Ω( σ ) (cid:16) n Y k =1 a i k · Φ( I ) (cid:17) == D D V (Φ) T V ( σ ) Ξ , Ξ E . (3.4) Proof.
We have D T V ( σ ) D V (Φ)Ξ , Ξ E = X I X J Φ( i σ − (1) , . . . , i σ − ( n ) ) n Y k =1 a / i σ − k ) n Y k =1 a / j k ×× (cid:28) η (cid:20) i σ − (1) . . . i σ − ( n ) i . . . i n (cid:21) , η (cid:20) j . . . j n j . . . j n (cid:21)(cid:29) . (3.5)A summand can be non-zero only if two basis elements in the brackets h· , ·i coincide. So i k = j k , i σ − k = j k . Therefore P I P J transforms to P I ∈ Ω( σ ) .Since I ∈ Ω( σ ), we can replace Φ( · ) by Φ( I ). Since J = I , the product Q k ( . . . ) Q k ( . . . ) replaces by Q k a i k .The expression for (cid:10) D V (Φ) T V ( σ ) Ξ , Ξ (cid:11) is similar to (3.5), we only mustreplace the boxed Φ( . . . ) by Φ( i , . . . , i n ). (cid:3) Lemma 3.3
D(cid:0) T V ( σ ) D V (Φ) (cid:1) D V (Θ)Ξ , Ξ E = D D V (Θ) (cid:0) T V ( σ ) D V (Φ) (cid:1) Ξ , Ξ E . (3.6) Proof.
To obtain the left hand side of (3.6), we must replace the boxedΦ( . . . ) in the calculation (3.5) byΦ( i σ − (1) , . . . , i σ − ( n ) ) Θ( i σ − (1) , . . . , i σ − ( n ) ) , in the right hand side byΘ( i , . . . , i n ) Φ( i σ − (1) , . . . , i σ − ( n ) ) . Since I ∈ Ω( σ ), we will get the same result in both sides. (cid:3) emma 3.4 D T V ( λ − ) T V ( σ ) D V (Φ) T V ( λ ) Ξ , Ξ E = D T V ( σ ) D V (Φ) Ξ , Ξ E . (3.7) Proof.
The left-hand side equals to X I X J Φ( i σ − λ (1) , . . . , i σ − λ ( n ) ) n Y k =1 a i λ − σ − λ ( k ) n Y k =1 a j k ×× ×× (cid:28) η (cid:20) i λ − σ − λ (1) . . . i λ − σ − λ ( n ) i . . . i n (cid:21) , η (cid:20) j . . . j n j . . . j n (cid:21)(cid:29) . As in proof of Lemma (3.4), we come to X I ∈ Ω( λ − σ − λ ) Φ( i σ − λ (1) , . . . , i σ − λ ( n ) ) n Y k =1 a i k . (we applied Lemma 3.2). Next, we pass to a new index of summation I ′ := ( i λ (1) , . . . , i λ ( n ) ) , I ′ ∈ Ω( σ ) . Keeping in mind Lemma 3.2 we get the desired expression in the right handside. (cid:3)
The statement a) of Lemma 3.1 follows from Lemmas 3.3 and 3.4.
Proof of the statement b) of Proposition 3.1.
We must verify theidentity T W ( λ ) D W (Ψ) Ξ = D V (Ψ) T V ( λ − ) Ξ . In the left hand-side we have X I Ψ( i λ − (1) , . . . , i λ − (1) ) Y a − / j λ − k η (cid:20) i . . . i n i λ − (1) . . . i λ − ( n ) (cid:21) . We pass to a new index of summation I ′ := ( i λ − (1) , . . . , i λ − ( n ) ) , and get the right hand side. (cid:3) The statement c) of Proposition 3.1 follows from b). ζ m .Lemma 3.5 Let H n acts in ( V ⊗ W ) ⊗ n as in Subsect. . Then (cid:10) ζ m Ξ , Ξ i = χ α,β, ( ζ ) . emma 3.6 Decompose the R -matrix (1.10) as R := Q + D, where Q : = −√ q X i = j,i =0 ,j =0 e ji ⊗ f ij ; D : = − X i< e ii ⊗ f ii + q X i> e ii ⊗ f ii + ( q − X i Denote a i := ( β − i , for i < α i , for i > D j ( j +1) are V -diagonal. Next, Q Ξ = −√ q X I =( i ,...,i n ): i = i Y k a / i k η (cid:20) i i i . . . i n i i i . . . i n (cid:21) The operators R , R , . . . can not change i in the first column of η [ · ], recallalso that such operators do not act on the second row of η [ · ]. Therefore allterms of R ( m − m . . . R Q Ξ have form c s η (cid:20) α s . . . α sn β s . . . β sn (cid:21) , where α s = β s .Inner product of such a term with Ξ is 0, i.e., (cid:10) R ( m − m . . . R Q Ξ , Ξ (cid:11) = 0 , hence (cid:10) R ( m − m . . . R R Ξ , Ξ (cid:11) = (cid:10) R ( m − m . . . R D Ξ , Ξ (cid:11) . We repeat the same argument for R , R , etc. and get the desired statement. (cid:3) The operator D ( m − m . . . D D is V -diagonal. Denote by δ [ . . . ] its eigen-values D ( m − m . . . D D η (cid:20) i . . . i m . . . i n j . . . j m . . . j n (cid:21) == δ ( i , . . . , i m ) η (cid:20) i . . . i m . . . i n j . . . j m . . . j n (cid:21) . emma 3.7 h ζ m Ξ , Ξ i = X i ,...,i m (cid:16) δ ( i , . . . , i m ) m Y k =1 a i k (cid:17) . (3.8) Proof. A straightforward summation gives X i ,...,i n (cid:16) δ ( i , . . . , i m ) n Y k =1 a i k (cid:17) We transform this expression as X i ,...,i m X i m +1 ,...,i n (cid:16) δ ( i , . . . , i m ) m Y k =1 a i k n Y k = m +1 a i k (cid:17) == X i ,...,i m (cid:16) δ ( i , . . . , i m ) m Y k =1 a i k (cid:17) × (cid:16)X i a i (cid:17) m − n . But under our conditions P a i = P β p + P α q = 1. (cid:3) Next, δ ( i , . . . , i n ) is non-zero iff i . . . i n . The next statement also isobvious: Lemma 3.8 Let a tuple I : i . . . i n consists of entries ( − ι u ) , . . . , ( − ι ) < with nonzero multiplicities µ , . . . , µ u and entries υ , . . . , υ v with nonzeromultiplicities ν , . . . , ν v . Then δ ( i , . . . , i n ) = ( − P k ( µ k − q P l ( ν l − ( q − u + v − . (3.9) Corollary 3.9 h ζ m Ξ , Ξ i = 1 q − X ϕ > , ϕ > , . . . ,ψ > , ψ > , · · · : P ϕ i + P ψ j = m ∞ Y i =1 (cid:16) ( − β i ) ϕ i (1 − q ) ε ( ϕ i ) (cid:17) ×× ∞ Y j =1 (cid:16) ( qα j ) ψ l (1 − q − ) ε ( ψ i ) (cid:17) , (3.10) where ε ( θ ) := ( , if θ = 0 ; , if θ > . Proof. We transform the expression (3.9) for δ ( . . . ) as1 q − Y k (cid:16) ( − µ k (1 − q ) (cid:17) · Y l (cid:16) q ν l (1 − q − ) (cid:17) . 19 straightforward summation in formula (3.8) gives an expression h ζ m Ξ , Ξ i = 1 q − X u,v : u + v > X ι < ··· <ι u ,υ < ··· <υ v X µ ,...,µ u ,ν ,...,ν v : P µ k + P ν l = m ×× u Y k =1 (cid:16) ( − β ι k ) µ k (1 − q ) (cid:17) v Y k =1 (cid:16) ( qα υ l ) ν l (1 − q − ) (cid:17) . We change a parametrization of the set of summation assuming ϕ ι k = µ k and ϕ i = 0 for all other i ; ψ υ l = ν l and ψ j = 0 for all other l ,and get (3.10), all new factors in the products in (3.10) are 1. (cid:3) Lemma 3.10 The generating function G ( z ) := 1 + ( q − (cid:16) z + X m > h ζ m Ξ , Ξ i z m (cid:17) is equal to G ( z ) := ∞ Y i =1 β i qz β i z ∞ Y j =1 − α j z − α j qz . (3.11) Proof. We represent the term ( q − z as( q − z = ( q − (cid:16)X j β j + X k α k (cid:17) z = X j ( − β j z )(1 − q ) + X k ( α k qz )(1 − q − ) . Applying (3.10) we come to the following expression for the generating function: G ( z ) = X ϕ > ,ϕ > ,...,ψ > ,ψ > ,... ∞ Y i =1 (cid:16) ( − β i z ) ϕ i (1 − q ) ε ( ϕ i ) (cid:17) ×× ∞ Y j =1 (cid:16) ( qα j z ) ψ j (1 − q − ) ε ( ψ j ) (cid:17) . It decomposes into a product ∞ Y i =1 (cid:16)X ϕ i ( − β i z ) ϕ i (1 − q ) ε ( ϕ i ) (cid:17) · ∞ Y j =1 (cid:16)X ψ j ( qα j z ) ψ j (1 − q − ) ε ( ψ j ) (cid:17) == ∞ Y i =1 (cid:16) − q ) (cid:0) (1 + β i z ) − − (cid:1)(cid:17) ∞ Y j =1 (cid:16) − q − ) (cid:0) (1 − qα j z ) − − (cid:1)(cid:17) . This equals to (3.11). (cid:3) roof of Lemma 3.5. We must verify the identity G ( z ) = 1 + ( q − z + X m > z m χ α,β, ( ζ m ) , see (1.7). We have G ( z ) = exp { ln G ( z ) } == exp nX i (cid:0) ln(1 + β i qz ) − ln(1 + β i z ) (cid:1) + X j (cid:0) ln(1 − α j z ) − ln(1 − α j qz ) (cid:1)o == exp nX i X k ( − k − k β ki ( q k − z k + X j X k k α kj ( q k − z k o == ∞ Y k =1 exp n(cid:0)X j α kj + ( − k − X j β ki (cid:1) · k ( q k − z k o . It remains to decompose exponentials. (cid:3) Obviously, R left i ( i +1) are contained in the alge-bra A V introduced in Subsect. 3.1. Therefore operators of the representation H n ( q ) are contained in A V . By Proposition 3.1.a, the matrix element h A Ξ , Ξ i isa trace on each H n . Therefore it is a trace on H ∞ ( q ). The properties (1.5)–(1.6)are obvious. After these remarks the statement a) of Theorem 1.5 follows fromLemma 3.5 and the statement b) from Proposition 3.1.b. References [1] Berezin F. A., Gelfand I. M. 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