Factorial multiparameter Hecke von Neumann algebras and representations of groups acting on right-angled buildings
FFactorial multiparameter Hecke von Neumann algebras andrepresentations of groups acting on right-angled buildings by Sven Raum and Adam Skalski
Abstract . We obtain a complete characterisation of factorial multiparameter Hecke von Neumannalgebras associated with right-angled Coxeter groups. Considering their (cid:96) p -convolution algebra ana-logues, we exhibit an interesting parameter dependence, contrasting phenomena observed earlier forgroup Banach algebras. Translated to Iwahori-Hecke von Neumann algebras, these results allow us todraw conclusions on spherical representation theory of groups acting on right-angled buildings, whichare in strong contrast to behaviour of spherical representations in the affine case. We also investigatecertain graph product representations of right-angled Coxeter groups and note that our von Neumannalgebraic structure results show that these are finite factor representations. Further classifying themup to unitary equivalence allows us to reveal high-dimensional Euclidean subspaces of the space ofextremal characters of right-angled Coxeter groups. Generic Hecke algebras are deformations of the group ring of a Coxeter group. Being intimatelyrelated to deep representation theoretic problems, they attracted great interest for spherical and affineCoxeter groups [KL79]. In recent decades, other Coxeter groups started to attract interest, driven bythe theory of buildings and by Kac-Moody groups acting on them [Rém12]. An operator algebraicperspective on Hecke algebras was created by Dymara who introduced Hecke von Neumann algebras N q ( W ) associated with a Coxeter system ( W, S ) and specific deformation parameters q ∈ R S > in orderto study cohomology of buildings [Dym06; DDJO07].Among non spherical and non affine Coxeter groups, the right-angled ones are particularly inter-esting. It was shown by Haglund and Paulin that buildings of arbitrary thickness associated withright-angled Coxeter groups exist [HP03]. Not only certain Kac-Moody groups act on right-angledbuildings, but also the buildings’ automorphism groups [Cap14] and universal groups with prescribedlocal action [DSS18; DS19] enrich the class of examples. From an operator algebraic point of view,right-angled Coxeter groups are interesting thanks to their rich combinatorial structure as graph prod-ucts [CF17], which makes a variety of operator algebraic tools available.The first structural result on Hecke von Neumann algebras was obtained by Garncarek, who charac-terised factorial single parameter Hecke von Neumann algebras of right-angled Coxeter groups [Gar16].Whether the multiparameter generalisation of his result holds remained unclear at the time and wasadvertised as an open problem in [Gar16, Question 2]. After further advances in the understandingof Hecke operator algebras [CSW19; Cas20], Caspers, Klisse and Larsen recently obtained in [CKL19]a factoriality result for some multiparameters, as a consequence of their simplicity and unique traceresults for Hecke C ∗ -algebras. However, they state in [CKL19, Remark 5.8 (a)] that Garncarek’s proof“does not trivially extend to the multiparameter case”. Here we do extend Garncarek’s work to themultiparameter case and thus obtain a complete solution to [Gar16, Question 2], at the same timeinitiating the study of the corresponding (cid:96) r -operator algebras. Our result is most conveniently formu-lated for parameters q ∈ ( , ] S , covering the general case thanks to standard considerations, based onKazhdan-Lusztig’s parameter reduction. Theorem A.
Let ( W, S ) be an irreducible, right-angled Coxeter system with at least three generators,let q ∈ ( , ] S and let r ∈ [ , ∞) . Write q w = q s ⋯ q s n for a reduced expression w = s ⋯ s n ∈ W . • If ∑ w ∈ W q r w = ∞ , then N r q ( W ) is a factor. MSC classification:
Keywords:
Hecke von Neumann algebra, II factor, right-angled Coxeter group, right-angled building, stronglytransitive action, spherical representation, character space a r X i v : . [ m a t h . OA ] O c t actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski • If ∑ w ∈ W q r w < ∞ , then N r q ( W ) = M ⊕ C p , where p is the projection onto the space of Heckeeigenvectors in (cid:96) r ( W ) and M is a factor. Including the (cid:96) r -convolution algebra analogues N r q ( W ) of Hecke von Neumann algebras allows us toexhibit the interesting parameter dependence, which is in contrast to simplicity results obtained sofar. Indeed, it is easy to see that the classical condition to have infinite (non-trivial) conjugacy classescharacterises factoriality of group convolution algebras on arbitrary (cid:96) r -spaces. Moreover, the articles[HP15; Phi19] show that also on a C ∗ -algebraic level, simplicity of a classical operator algebra associatedwith a group implies simplicity of its (cid:96) r -analogues. Our result shows that for Hecke operator algebras asimilar implication holds no longer true. The remark is due that some simplicity results for the norm-closed (cid:96) r -analogues of Hecke operator algebras can be obtained from existing literature. Combiningthe averaging operators introduced in [CKL19] with the interpolating methods employed in [Phi19],it follows directly that for every Hecke C ∗ -algebra C ∗ q ( W ) for which [CKL19] proves simplicity, allnorm-closed (cid:96) r -analogues are simple for r from a neighbourhood of . The crucial difference to [Phi19],preventing further conclusions, is the lack of a good norm-estimate for the averaging operators on (cid:96) ( W ) .It must also be mentioned that the infinite dimensional von Neumann factors arising in Theorem Aare non-injective by [CKL19, Theorem 6.2]. Furthermore, an observation made already in [Gar16,Section 6] (compare also with [CKL19, Section 5.4]) is that Dykema’s free probability results [Dyk93]apply in the special case of free products W = Z / Z ∗ n and yield there the von Neumann algebraicstatement of our Theorem A.It is fair to say that recent literature treated Hecke operator algebras as analytic-combinatorialobjects. Concerning our main methods, the present work is no different. This is not the case whenapplications are concerned. The origin of Hecke operator algebras is geometric and lies in the Singerconjecture on cohomology of buildings. But already Garncarek has to state that “it turns out, althoughthe centers of N q ( W ) can be nontrivial, they contribute nothing new in the subject of decomposingthe weighted cohomology of W ” [Gar16, p.1203]. Subsequent work on Hecke operator algebras makesno different claim, and also our work does not create new insight into weighted cohomology. However,connections between Hecke operator algebras and the representation theory of groups acting on build-ings have so far not appeared in the literature. A classical result in representation theory of p-adicreductive groups going back to Iwahori and Matsumoto [IM65] says that generic Hecke algebras areisomorphic with double coset Hecke algebras associated with Iwahori subgroups. By definition, an Iwa-hori subgroup of a p-adic reductive group is the stabiliser of a chamber in the associated Bruhat-Titsbuilding. This identification of Hecke algebras continues to hold for groups acting strongly transitivelyon arbitrary buildings. We are hence able to deduce information about the spherical representationtheory of certain groups acting on right-angled buildings from (the Hilbert space case of) Theorem A.The most precise way to do so, is to describe a decomposition of the quasi-regular representation asso-ciated with an Iwahori subgroup. It is unitarily equivalent to a restriction of the regular representation,since Iwahori subgroups are compact. The next corollary applies to several classes of groups mentionedbefore, such as certain completed Kac-Moody groups [RR06; Rém12], the full group of type preservingautomorphism of right-angled buildings [Cap14] and certain universal groups with prescribed localaction [DSS18; DS19]. Corollary B.
Let G ≤ Aut ( X ) be a closed subgroup acting strongly transitively and type preservingon a right-angled irreducible thick building X of type ( W, S ) and rank at least three. Denote by ( d s ) s ∈ S the thickness of X and let B ≤ G be the stabiliser of a chamber in X and let K ≤ G be the stabiliser ofa vertex in X . We write d w = d s ⋯ d s n for a reduced expression w = s ⋯ s n ∈ W . • If ∑ w ∈ W d − w = ∞ , then the quasi-regular representation λ G,B on (cid:96) ( G / B ) is a type II ∞ factorrepresentation. • If ∑ w ∈ W d − w < ∞ , then λ G,B is the direct sum of a type II ∞ factor representation and an infinitedimensional irreducible representation. by Sven Raum and Adam Skalski • λ G,K is a type II ∞ factor representation. This result provides a complete description of square integrable Iwahori-spherical representations.The fact that there is an irreducible direct summand of λ G,B if ∑ w ∈ W d − w < ∞ follows from theexistence of the Steinberg representations, already described by Matsumoto in [Mat69, Number 1.3].Factoriality of its orthogonal complement is in stark contrast to previously observed behaviour ofspherical representation theory. In the classical cases, such as groups acting strongly transitively andtype preserving on trees [Car73; FN91] and more generally on affine buildings [Mat77; CC15], theIwahori-Hecke algebra associated with B is a finitely generated module over the abelian sphericalHecke algebra, which is associated with the maximal compact subgroup K . It was already shown in[Léc10, Theorem 3.5] (see also [APV13] and [Mon20, Corollary 2]) that the spherical Hecke algebra inthe setting of our Corollary B cannot be abelian. The decomposition of the quasi-regular representationwe provide, however, goes far beyond the statement of the spherical Hecke algebra being non-abelian,and similar examples were not obtained before. In particular, it follows already from [Suz17] (see also[Rau19, Theorem G] whose proof is not correct as stated, but the result can be recovered based onSuzuki’s article) that there are groups acting on trees whose regular representation is factorial, but thegroup action is not (and cannot be) strongly transitive.In the second part of this article, we focus on unitary representation theory of right-angled Coxetergroups. In [Dav08, Notes 19.2] and [CKL19, Proposition 4.2] it was observed that generic Heckealgebras associated with right-angled Coxeter groups are isomorphic with the respective group algebras.A similar phenomenon is known in the spherical case [Bou02, Chapter IV, Exercise 27]. We put thisobservation into a systematic context of graph products of representations. Thanks to Theorem Awe can single out a family λ a , a ∈ (− , ) S of finite factor representations arising from graph productrepresentations of an irreducible, right-angled Coxeter system on at least three generators. Our secondmain result shows that these representations are pairwise not unitarily equivalent, even though theysometimes generate the same Hecke von Neumann algebra. This is an interesting phenomenon in viewof the open classification problems for Hecke operator algebras and it has non-trivial consequences forthe character space of right-angled Coxeter groups. Theorem C.
Let ( W, S ) be an irreducible, right-angled Coxeter system with at least three generators.There are finite factor representations λ a , a ∈ (− , ) S of W that are pairwise unitarily inequivalentand such that N q ( W ) = λ a ( W ) ′′ ⊕ C p for q s = −∣ a s ∣√ − a s , where p is a projection of rank at most one,which vanishes if and only if ∑ w ∈ W q w = ∞ . Combining the continuous parameter dependence obtained from the construction as graph prod-uct representations, Theorem C implies the existence of high-dimensional Euclidean subspaces of thecharacter space of right-angled Coxeter groups. This result is in contrast to recent classification resultsfor traces on group C ∗ -algebras obtained for example in [Bek07; PT16; CP13; BH19b; Bek20; BF20;BBHP20]. While graph products of groups are expected to have complicated trace spaces, their precisetopological structure remains unclear. Corollary D.
Let ( W, S ) be an irreducible, right-angled Coxeter system with at least three generators.There is an ∣ S ∣ -dimensional Euclidean subspace of the extremal points of Char ( W ) whose interiorcontains the regular trace. This article contains four sections. After the introduction, we collect some preliminaries on thevarious topics we treat. In Section 3, we prove our first main Theorem A and its Corollary B. InSection 4, we introduce the graph products of representations and prove Theorem C and Corollary D.
Acknowledgements
S.R. was supported by the Swedish Research Council through grant number 2018-04243. A.S. waspartially supported by the National Science Center (NCN) grant no. 2014/14/E/ST1/00525.3actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski
The authors are deeply grateful to Pierre-Emmanuel Caprace for pointing out that the factorrepresentation in Corollary B cannot be finite, and for clarifying remarks on the representation theoryof reductive groups. His remarks also led us to state the factoriality result for spherical representations.We thank Mateusz Wasilewski for useful comments improving the exposition of our work. We alsothank Pierre-Emmanuel Caprace and Bertrand Rémy for pointing out the references [APV13] and[Léc10].
Standard resources for this material are [DE14] and the appendices of [BHV08]. Every locally compactgroup G admits a left invariant Radon measure µ , which is unique up to a positive scalar. The modularfunction of G is the unique group homomorphism ∆ ∶ G → R > satisfying ∆ ( x ) µ ( B ) = µ ( Bx ) for allmeasurable B and all x ∈ G .The space of continuous and compactly supported functions C c ( G ) becomes a *-algebra whenequipped with the product f ∗ g ( x ) = ∫ G f ( y ) g ( y − x ) d µ ( y ) and the involution f ∗ ( x ) = ∆ ( x − ) f ( x − ) .The convolution product of C c ( G ) extends to a *-representation C c ( G ) → B( L ( G )) . Given a compactopen subgroup K ≤ G , its (coset) Hecke algebra is the *-subalgebra C [ G, K ] ⊂ C c ( G ) of K -biinvariantfunctions. The Hecke algebra can be described as C [ G, K ] = p K C c ( G ) p K where p K = K is theindicator function of K ; we shall always normalise the Haar mesaure of G on a suitable compact opensubgroup K .The left regular representation is the the unique group homomorphism λ ∶ G → U ( L ( G )) satisfying λ x ( f )( y ) = f ( x − y ) for all f ∈ C c ( G ) ⊂ L ( G ) and all x, y ∈ G . The group von Neumann algebra of G is the double commutant L ( G ) = λ ( G ) ′′ = C c ( G ) SOT ,where SOT denotes the strong operator topology, that is the topology of pointwise convergence. Givena closed subgroup H ≤ G , we denote by λ G,H ∶ G → L ( G / H ) the quasi-regular representation . If H ≤ G is open, G / H is discrete and λ G,H is the permutation representation of the left translation action on G / H . In particular, it has no finite dimensional subrepresentations if G / H is infinite. For a compactopen subgroup K ≤ G , the quasi-regular representation λ G,K is unitarily equivalent to the restrictionof λ to the subrepresentation generated by K -fixed vectors. The Hecke von Neumann algebra of thepair K ≤ G is L ( G, K ) = p K L ( G ) p K = C [ G, K ] SOT ⊂ B( (cid:96) ( K / G )) .Both the group von Neumann algebra and the Hecke von Neumann algebra admit counterparts R ( G ) and R ( G, K ) constructed from the right regular representation of G , which is defined by ρ x ( f )( y ) = ∆ G ( x ) f ( yx ) for all f ∈ C c ( G ) ⊂ L ( G ) and all x, y ∈ G . Since λ and ρ are unitarily equivalent, wehave L ( G ) ≅ R ( G ) and L ( G, K ) ≅ R ( G, K ) .A unitary representation π ∶ G → U ( H ) is a subrepresentation of λ ⊕ κ for some cardinal κ if andonly if it is square integrable in the sense that for a dense set of ξ ∈ H the function g ↦ ⟨ π ( g ) ξ, ξ ⟩ issquare integrable with respect to the Haar measure of G . See [Dix96, Chapter 14] or [Rie69, Theorem4.6]. In order to avoid a clash of terminology, we prefer the term “square integrable” to “discrete series”here, since the latter usually makes the additional assumption of irreducibility in the representationtheory of reductive groups. 4actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski The books [Hum90] and [Dav08] both describe Coxeter groups. A
Coxeter matrix over a non-emptyset S is a symmetric S × S matrix whose diagonal entries equal one and whose off diagonal entries liein N ≥ ∪ {∞} . Let M = ( m st ) s,t ∈ S be a Coxeter matrix. The Coxeter system ( W, S ) associated with M is given by the group W = ⟨ S ∣ ∀ s, t ∈ S ∶ e = ( st ) m st ⟩ together with its generating set S . The Coxeter matrix and its Coxeter system are called right-angled if m st ∈ { , ∞} for all different s, t ∈ S . They are irreducible if the Coxeter matrix is not the Kroneckertensor product of two non-trivial Coxeter matrices. Equivalently, there is no partition S = S ⊔ S suchthat W = ⟨ S ⟩ ⊕ ⟨ S ⟩ .Given w ∈ W we denote by ∣ w ∣ its word length with respect to the generating set S . A word w withletters in S is called reduced if it has ∣ w ∣ many letters. The s -length ∣ w ∣ s of w ∈ W is the number ofoccurrences of s in any reduced word representing w . Tits’ solution to the word-problem in Coxetergroups [Tit69, Théorème 3] provides normal forms for elements in W and shows that the s -length iswell defined. Given a subgroup D = ⟨ T ⟩ for T ⊂ S , we call a double coset DwD ⊂ W non-degenerate ifit is not equal to a simple coset. The book [Hum90] introduces generic Hecke algebras. Also [Dav08] introduces Hecke algebras anddiscusses their von Neumann algebraic completions. Given a Coxeter system ( W, S ) , we denote its real deformation parameters by R ( W,S )> = { q ∶= ( q s ) s ∈ S ∈ R S > ∣ ∀ s, t ∈ S ∶ (∃ w ∈ W ∶ wsw − = t (cid:212)⇒ q s = q t )} .Frequently, the expression p s = q s − √ q s will be used. If ( W, S ) is right-angled, Tits’ solution to the word-problem implies that R ( W,S )> = R S > .Indeed, if s, t, s , . . . , s n ∈ S satisfy s ⋯ s n ss n ⋯ s = t , then comparing the length of both sides, it followsthat [ s, s i ] = e for all i ∈ { , . . . , n } and thus s = t . For q ∈ R ( W,S )> and a reduced word w = s ⋯ s n ∈ W we write q w = q s ⋯ q s n .The (generic) Hecke algebra associated with ( W, S ) and the multiparameter q ∈ R ( W,S )> is the*-algebra with the presentation C q [ W ] = ⟨( T ( q ) s ) s ∈ S ∣∀ s, t ∈ S, m st < ∞ ∶ T ( q ) s T ( q ) t ⋯ = T ( q ) t T ( q ) s ⋯ ( m st many factors) , ( T ( q ) s − q s )( T ( q ) s + q − s ) = , ( T ( q ) s ) ∗ = T ( q ) s ⟩ .For a reduced expression w = s ⋯ s n ∈ W , we write T ( q ) w = T ( q ) s ⋯ T ( q ) s n . One can check that this doesnot depend on the choice of the reduced expression. In particular, ( T ( q ) w ) ∗ = T ( q ) w − holds for all w ∈ W .Further, for all w, w ′ ∈ W satisfying ∣ ww ′ ∣ = ∣ w ∣ + ∣ w ′ ∣ we have T ( q ) ww ′ = T ( q ) w T ( q ) w ′ .For every r ∈ [ , ∞) , the algebra C q [ W ] admits a representation by bounded operators on (cid:96) r ( W ) satisfying the following formula on its generators. π r q ( T ( q ) s ) δ w = ⎧⎪⎪⎨⎪⎪⎩ δ sw if ∣ sw ∣ > ∣ w ∣ ,δ sw + p s δ w if ∣ sw ∣ < ∣ w ∣ .5actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski We define the Hecke (cid:96) r -convolution algebra by N r q ( W ) = π r q ( C q [ W ]) ′′ .Note that the vector δ e ∈ (cid:96) r ( W ) is cyclic and separating, as a short calculation using J ∶ (cid:96) r ( W ) —→ (cid:96) r ( W ) ∶ δ w ↦ δ w − shows. This allows us to study elements x ∈ N r q ( W ) by means of their image ˆ x = xδ e ∈ (cid:96) r ( W ) and shows injectivity of π r q . We will henceforth suppress the representation π r q in ournotation, identifying T ( q ) w with its image in B( (cid:96) r ( W )) .In the Hilbert space case r = , the algebra N q ( W ) = N q ( W ) is the multiparameter Hecke vonNeumann algebra associated with ( W, S ) and q ∈ R ( W,S )> . The representation π q is a *-representationand it arises from the GNS-construction with respect to the tracial state T ( q ) w ↦ ⟨ T ( q ) w δ e , δ e ⟩ = δ w,e on C q [ W ] .The following parameter reduction is well known from Kazhdan-Lusztig’s work [KL79]. See also[CKL19, Proposition 4.7] for the proof of an operator algebraic version for r = . The statementcontinues to hold in the (cid:96) r -setting with the same proof. Proposition 2.1.
Let ( W, S ) be a right-angled Coxeter system, let q ∈ R S > , ( (cid:15) s ) s ∈ S ∈ {− , } S and r ∈ [ , ∞) . Define q ′ = ( q (cid:15) s s ) s ∈ S . Then the map C q [ W ] —→ C q ′ [ W ] ∶ T ( q ) s ↦ (cid:15) s T ( q ′ ) s is well defined and extends to an isometric isomorphism N r q ( W ) ≅ N r q ′ ( W ) . The following description of Hecke eigenvectors was first observed in [DDJO07, Section 5]. See also[Gar16, Proof of Theorem 5.3].
Proposition 2.2.
Let ( W, S ) be a right-angled Coxeter system, let q ∈ ( , ] S and r ∈ [ , ∞) . Then theHecke operators ( T ( q ) s ) s ∈ S have a common eigenvector in (cid:96) r ( W ) if and only if ∑ w ∈ W q r w < ∞ . In thiscase, every common eigenvector of Hecke operators is a multiple of ξ = ∑ w ∈ W q w δ w and the projectiononto C ξ lies in the centre of N r q ( W ) . The eigenvalues are given by T ( q ) s p = √ q s p , s ∈ S . The standard reference for buildings is [AB08]. We follow Garrett’s presentation from [Gar97] as itleads directly to the comparison of generic Hecke algebras and Iwahori-Hecke algebras. A chambercomplex is a simplicial complex X such that all simplices of X are contained in a maximal simplex andsuch that for every pair x, y of maximal simplices of X there is a chain x = , x , . . . , x n = y of adjacentmaximal simplices in X . Its maximal simplices are called chambers . They will usually be denoted by C . A chamber complex is thin if each facet of a chamber is the facet of exactly two chambers, and itis called thick if each facet of a chamber is the facet of at least three chambers. An important exampleof a thin chamber complex is the Coxeter complex Σ ( W, S ) associated with a Coxeter system ( W, S ) .Its simplices are indexed by the cosets W /⟨ T ⟩ , where T ⊂ S is a proper subset. The face relationis determined by opposite inclusion. In particular, the singletons w ⟨∅⟩ = { w } represent the maximalsimplices of Σ ( W, S ) .A system of apartments in a chamber complex X is a set A of thin chamber subcomplexes suchthat• for each pair of simplices x, y of X there is A ∈ A such that x, y ∈ A , and• if A, A ′ ∈ A contain a common chamber C and a simplex x , then there is an isomorphism ofchamber complexes φ ∶ A → A ′ that fixes C and x point-wise.6actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski A thick building is a thick chamber complex that admits a system of apartments. One can prove thatfor a thick building X there is a unique maximal apartment system. So if X is thick, as we assumefrom now on, it makes sense to speak of an apartment of X without reference to a specific system ofapartments. There is a Coxeter system ( W, S ) such that every apartment of X is chamber isomorphicwith Σ ( W, S ) . The choice of a fundamental chamber C in X and a compatible labelling of its facetsby elements of S makes these isomorphisms unique and introduces a labelling of all facets of chambersin X . The Coxeter system ( W, S ) is called the type of X . In particular, a right-angled building is abuilding whose Coxeter system is right-angled, and an irreducible building is a building whose Coxetersystem is irreducible (the latter property admits also an intrinsic description). The rank of X is bydefinition the rank of ( W, S ) , which is equal to the number of facets of any chamber in X . The thickness of X is the tuple ( d s ) s ∈ S where d s is the cardinality of the set of chambers containing the fundamentalchamber’s facet labelled by s . The building is called locally finite if all d s , s ∈ S are finite.Given a locally finite, thick building X , its group of simplicial automorphisms Aut ( X ) carriesthe topology of point-wise convergence on simplices, which turns Aut ( X ) into a totally disconnected,locally compact group. A subgroup G ≤ Aut ( X ) is called type preserving , if it leaves invariant anylabelling of facets of chambers in X obtained from the choice of a labelled fundamental chamber. Asubgroup G ≤ Aut ( X ) is called strongly transitive if it acts transitively on pairs ( A, C ) where A is anapartment of X and C is a chamber in A . Given a closed, strongly transitive subgroup G ≤ Aut ( X ) , an Iwahori subgroup of G , denoted below by B , is the stabiliser of a chamber in X . Every Iwahori subgroupis compact and open and it is unique up to conjugation by elements from G . Iwahori subgroups playthe role of the group B in Tits’ ( B, N ) -pairs, which can be used to construct buildings from groups.If ( W, S ) is the type of X , then the Bruhat decomposition of a closed, strongly transitive and typepreserving subgroup G ≤ Aut ( X ) provides a bijection W → B / G / B assigning to w ∈ W its Bruhat cell
BwB . Note that this identification uses the fact that W can be viewed as the group of type preservingautomorphisms of an apartment of X . The Iwahori-Hecke algebra of G is the algebra C [ G, B ] of B -biinvariant and compactly supported functions on G equipped with the convolution product of C c ( G ) .We normalise the Haar measure of G on some (or equivalently any) Iwahori subgroup. A unitaryrepresentation π of G is called Iwahori-spherical if it has a non-zero fixed vector under some Iwahorisubgroup of G .The following identification of Hecke algebras goes back to Iwahori and Matsumoto [IM65]. Werefer to [Gar97, Section 6.2] for a modern presentation, and for the convenience of the reader give ashort proof, showing that the scaling factors are chosen correctly. Theorem 2.3.
Let X be a locally finite, thick building of type ( W, S ) and G ≤ Aut ( X ) be a closed,strongly transitive and type preserving subgroup. Let B ≤ G be an Iwahori subgroup of G and denoteby ( d s ) s ∈ S the thickness of X . Let q s = d − s , for s ∈ S . Then C q ( W ) ≅ C [ G, B ] as *-algebras, with anisomorphism given by the map T ( q ) w ↦ (− ) ∣ w ∣ √ q w BwB .Proof.
In [Gar97, Section 6.2], it is shown that C [ G, B ] is presented as the algebra with relations C [ G, B ] = ⟨( BwB ) w ∈ W ∣ BsB BwB = BswB if ∣ sw ∣ > ∣ w ∣ , BsB = ( d s − ) BsB + d s ⟩ .Using the first relation, we see that for s ≠ t in S the Artin relation BsB BtB ⋯ = BtB BsB ⋯ ( m st factors ) is satisfied, where m st denotes the order of st in W . The second relation implies that ( − √ d s BsB ) = d s − d s BsB + = − p s √ q s BsB + ,which is equivalent to the Hecke relation of C q ( W ) . So there is an algebra homomorphism ϕ ∶ C q ( W ) → C [ G, B ] satisfying ϕ ( T ( q ) s ) = (− )√ q s BsB for all s ∈ S . It satisfies ϕ ( T ( q ) w ) = (− ) ∣ w ∣ √ q w BwB for7actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski all w ∈ W by the definition of T ( q ) w and the defining relations of C [ G, B ] . An inverse for ϕ existssince (− ) ∣ w ∣ √ d w T ( q ) w satisfy the relations characterising C [ G, B ] . So ϕ is an algebra isomorphism.We conclude by pointing out that it is a *-isomorphism thanks to the fact that ( T ( q ) w ) ∗ = T ( q ) w − and ∗ BwB = Bw − B . In this section, we will prove Theorem A and Corollary B. Our strategy to prove Theorem A closelyfollows [Gar16], with the key new result being our Lemma 3.2.The proof of the following lemma is standard. Its single parameter (Hilbert space) version alreadyappeared as [Gar16, Lemma 5.1]. Recall that ˆ x = xδ e ∈ (cid:96) r ( W ) for each x ∈ N r q ( W ) . Lemma 3.1.
Let ( W, S ) be a right-angled Coxeter system and q ∈ ( , ] S . Fix s ∈ S and x ∈ N r q ( W ) .Then x commutes with T ( q ) s if and only if for all w ∈ W satisfying ∣ sws ∣ = ∣ w ∣ + we have ˆ x sw = ˆ x ws ˆ x sws = ˆ x w + p s ˆ x sw . The key new result in our treatment of the multiparameter case compared to [Gar16] is the followinggeneralisation of [Gar16, Proposition 5.2].
Lemma 3.2.
Let ( W, S ) be a right-angled Coxeter system, let q ∈ ( , ] S and let r ∈ [ , ∞) . Let s, t ∈ S be non-commuting elements and write D = ⟨ s, t ⟩ . Assume that x ∈ N r q ( W ) commutes with T ( q ) s and T ( q ) t . Then for every non-degenerate double coset DwD ⊂ W with shortest element w , we have ˆ x dwd ′ = ˆ x w ⋅ ( q dwd ′ q w ) for all d, d ′ ∈ D .Proof. Since
DwD is non-degenerate, at least one of the elements s, t does not commute with w . Wemay without loss of generality assume that this is t . Define a function f ∶ D × D → C by f ( d, d ′ ) = ˆ x d − wd ′ .Then we have to show that f ( d, d ′ ) = ˆ x w ⋅ ( q d − wd ′ q w ) .We will first collect several algebraic relations between values of f , obtained by applying Lemma 3.1.Given d, d ′ ∈ D that do not end in t we have ∣ td − wd ′ t ∣ = ∣ d − wd ′ ∣ + , whence f ( d, d ′ t ) = f ( dt, d ′ ) (3.1) f ( d, d ′ ) = f ( dt, d ′ t ) − p t f ( dt, d ′ ) .(3.2)If s does not commute with w , we similarly find that for all d, d ′ ∈ D that do not end in s , we have f ( d, d ′ s ) = f ( ds, d ′ ) (3.3) f ( d, d ′ ) = f ( ds, d ′ s ) − p s f ( ds, d ′ ) .(3.4)If s does commute with w , these statements do hold for d, d ′ ∈ D that end in t and additionally for thechoice d = e and d ′ ∈ D that does not start or end with s , as in these cases ∣ sd − wd ′ s ∣ = ∣ d − wd ′ ∣ + continues to hold. Claim 1.
The equality f ( d, d ′ ts ) = f ( dts, d ′ ) holds for all d, d ′ ∈ D that do not end with t . by Sven Raum and Adam Skalski Proof of the claim.
We fix a pair of elements d, d ′ ∈ D that do not end in t . Write d n = dtstst ⋯ with n letters attached, and similarly write d ′ n = d ′ tstst ⋯ with n letters attached. We can apply the relations(3.2) and (3.4) to obtain f ( d, d ′ ) = f ( dt, d ′ t ) − p t f ( dt, d ′ )= f ( dts, d ′ ts ) − p s f ( dts, d ′ t ) − p t f ( dt, d ′ )= f ( d n , d ′ n ) − n − ∑ k = p s f ( d k ts, d ′ k t ) + p t f ( d k t, d ′ k ) .Since f vanishes at infinity, we infer that the partial sums ∑ n − k = p s f ( d k ts, d ′ k t )+ p t f ( d k t, d ′ k ) convergeand we obtain the expression f ( d, d ′ ) = − ∞ ∑ k = p s f ( d k ts, d ′ k t ) + p t f ( d k t, d ′ k ) .In order to prove our claim, it now suffices to see that f ( d ( ts ) k + t, d ′ ( ts ) k ) = f ( d ( ts ) k t, d ′ ( ts ) k + ) and f ( d ( ts ) k + , d ′ ( ts ) k t ) = f ( d ( ts ) k + , d ′ ( ts ) k + t ) hold for all k ∈ N . These relations follow from equations(3.1) and (3.3). Claim 2. If s and w do not commute, then f ( d, d ′ st ) = f ( dst, d ′ ) holds for all d, d ′ ∈ D that do notend with s . If s and w commute, then f ( d, d ′ st ) = f ( dst, d ′ ) holds for all d, d ′ ∈ D that end with t aswell as for d = e under the additional condition that d ′ ∈ W neither starts nor ends with s .Proof of the claim. The proof of this claim follows exactly the same way as the previous one, invokingequations (3.2) and (3.4) while respecting the special case where s and w commute.Let us define two functions g , g ∶ N → C by the formulas g ( n ) = f (( ts ) n , e ) g ( n ) = f (( ts ) n t, e ) .Then g ( n + ) = g ( n ) + p s g ( n + ) + p t g ( n ) for n ∈ N ,(3.5)since an application of equations (3.2) and (3.4), and then of Claim 1 and equation (3.1) shows that f (( ts ) n , e ) = f (( ts ) n t, t ) − p t f (( ts ) n t, e )= f (( ts ) n ts, ts ) − p s f (( ts ) n ts, t ) − p t f (( ts ) n t, e )= f (( ts ) n + , e ) − p s f (( ts ) n + t, e ) − p t f (( ts ) n t, e ) .Similarly, the recurrence g ( n + ) = g ( n ) + p t g ( n + ) + p s g ( n + ) for n ∈ N (3.6)follows from the calculation f (( ts ) n t, e ) = f (( ts ) n ts, s ) − p s f (( ts ) n ts, e )= f (( ts ) n tst, st ) − p t f (( ts ) n tst, s ) − p s f (( ts ) n ts, e )= f (( ts ) n + t, e ) − p t f (( ts ) n + , e ) − p s f (( ts ) n + , e ) .The system of recurrence relations (3.5)–(3.6) associated with g , g can be transformed into a linearsystem of four relations of with constant coefficients. It follows that it has a system of four fundamental9actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski solutions. Considering the solutions to the difference equation appearing in [Gar16, Proposition 5.2](and correcting the typo in there), one guesses the following solutions and verifies in elementarycalculations that they indeed obey the recurrences. ˜ g ( n ) = q n s q n t (First fundamental solution) ˜ g ( n ) = q n s q n + t ˜ g ′ ( n ) = (− ) n q − n s q n t (Second fundamental solution) ˜ g ′ ( n ) = (− ) n q − n s q n + t ˜ g ′′ ( n ) = (− ) n q n s q − n t (Third fundamental solution) ˜ g ′′ ( n ) = (− ) n + q n s q − n + t ˜ g ′′′ ( n ) = q − n s q − n t (Fourth fundamental solution) ˜ g ′′′ ( n ) = − q − n s q − n + t .We exemplify the necessary calculations by checking equation (3.5) for the first fundamental solution ( ˜ g , ˜ g ) . Dividing both sides of (3.5) by ˜ g ( n ) , it remains to match the term q s q t with + p s q s q t + p t q t = + ( q s − ) q t + ( q t − ) = q s q t .So the equation is indeed satisfied.Because q s , q t ∈ ( , ] , it follows that scalar multiples of the first fundamental solution are the onlysolutions to the recurrence that vanish at infinity. So we find that f (( ts ) n , e ) = g ( n ) = g ( ) ˜ g ( n ) = f ( e, e ) q n s q n t (3.7)and similarly f (( ts ) n t, e ) = g ( n ) = g ( ) ˜ g ( n ) = f ( e, e ) q n s q n + t .We will next investigate the pairs (( st ) n , e ) and (( st ) n s, e ) in a similar fashion, carefully taking careof possible commutation between w and s . Let us define functions h , h ∶ N → C by h ( n ) = f (( st ) n , e ) h ( n ) = f (( st ) n s, e ) .If s and w do not commute, then we can derive and solve recurrence relations as before and therebyfind that f (( st ) n , e ) = q n s q n t f (( st ) n s, e ) = q n + s q n t .If s and w commute, we only obtain the relations h ( n + ) = h ( n ) + p t h ( n + ) + p s h ( n ) for n ∈ N h ( n + ) = h ( n ) + p s h ( n + ) + p t h ( n + ) for n ∈ N ≥ .10actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski The same method as before applies, showing that for n ∈ N ≥ we have h ( n ) = h ( ) q n − s q n − t h ( n ) = h ( ) q n s q n − t .We note that equations (3.1) and (3.3) combined with Claim 1 imply that f ( st, e ) = f ( s, t ) = f ( e, ts ) = f ( ts, e ) .Using this together with formula (3.7) shows that h ( ) = f ( st, e ) = f ( ts, e ) = f ( e, e ) q s q t .This implies that f (( st ) n , e ) = h ( n ) = h ( ) q n − s q n − t = f ( e, e ) q n s q n t for n ∈ N ≥ and similarly f (( st ) n s, e ) = h ( n ) = h ( ) q n s q n − t = f ( e, e ) q n + s q n t for n ∈ N ≥ .Since h ( ) = f ( e, e ) holds by definition, it remains to find the value of h ( ) = f ( s, e ) . Plugging into the first recurrence relation, we obtain f ( e, e ) q s q t = f ( e, e )( + p t q s q t ) + p s f ( s, e ) and thus f ( s, e ) = p − s f ( e, e )( q s q t − − p t q s q t )= p − s f ( e, e )( q s q t − − q s ( q t − ))= p − s f ( e, e )( q s − )= f ( e, e ) q s .Summarising our results so far, we have seen that f ( d, e ) = f ( e, e ) q d = f ( e, e ) ( q d − w q w ) holds for all d ∈ D . In the remainder of the proof, we will reduce all considerations about the values of f to these cases.Since the relations described in (3.1) and (3.3) as well as in Claims 1 and 2 do not change the value of q d − wd ′ , we see that it remains to determine the values of f on the following pairs of elements: (( st ) n s, s ) for n ∈ N , (( ts ) n t, t ) for n ∈ N , (( st ) n , t ) for n ∈ N ≥ , (( ts ) n , s ) for n ∈ N ≥ . Treating the exception first, we observe that if s and w commute, then (( st ) n s ) − ws = ( st ) n sws =( st ) n w and hence f (( st ) n s, s ) = f (( st ) n , e ) . If s and w do not commute, we obtain f (( st ) n s, s ) = f (( st ) n , e ) + p s f (( st ) n s, e )= f ( e, e ) q n s q n t ( + p s q s )= f ( e, e ) q n s q n t ( + q s − )= f ( e, e ) q n + s q n t .Similar calculations determine the value of f on the pairs (( st ) n , t ) , n ∈ N ≥ as well as (( ts ) n t, t ) , n ∈ N and (( ts ) n , s ) , n ∈ N ≥ , thereby finishing the proof of the lemma.11actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski In order to complete Lemma 3.2 to a proof of Theorem A, we will make use of the following combi-natorial result proven by Garncarek.
Proposition 3.3 ([Gar16, Proposition 3.4]).
Let ( W, S ) be an irreducible, right-angled Coxetersystem with at least three generators. Denote by Γ the graph whose vertex set is W and for which w, w ′ ∈ W are adjacent if and only if there are s, t ∈ S such that for D = ⟨ s, t ⟩ , we have DwD = Dw ′ D and this coset is non-degenerate. Then all elements of W ∖( S ∪{ e }) lie in the same connected componentof Γ . Having Lemma 3.2 at hand, the proof of [Gar16, Theorem 5.3] carries over to the multiparametercase.
Proof of Theorem A.
Assume that x lies in the centre of N r q ( W ) . Let v, w ∈ W and assume thatthere are s, t ∈ S such that for D = ⟨ s, t ⟩ , the cosets DvD and
DwD are equal to each other andnon-degenerate. Then Lemma 3.2 implies that ˆ x w q v = ˆ x v q w .By Proposition 3.3 it follows that there is c ∈ C such that ˆ x w = cq w for all w ∈ W ∖ ( S ∪ { e }) .(3.8)Let s ∈ S and take t ∈ S that does not commute with s . Then Lemma 3.1 implies that ˆ x s = ˆ x tst − p t ˆ x ts = cq t q s − p t cq t q s = c ( q t q s − ( q t − ) q s )= cq s .This shows that (3.8) holds for all w ∈ W ∖ { e } . So there is d ∈ C such that ˆ x w = dδ e + cq w holds forall w ∈ W . If ∑ w ∈ W q r w = ∞ holds, then we must necessarily have c = so that x ∈ C follows, whichfinishes the proof in this case. If on the other hand ∑ w ∈ W q r w < ∞ , then ˆ x is a linear combination of δ e and ξ = ∑ w ∈ W q w δ w ∈ (cid:96) r ( W ) . Since the projection onto C ξ lies in N r q ( W ) by Proposition 2.2, theproof is also finished in this case.We will next use the relation between representations of groups acting on right-angled buildings andgeneric Hecke von Neumann algebras described in Section 2.4. Let us first observe that the identificationof Hecke algebras made in Theorem 2.3 extends to an identification of Hecke von Neumann algebras,before we prove Corollary B. Theorem 3.4.
Let X be a locally finite, thick building of type ( W, S ) and G ≤ Aut ( X ) be a closed,strongly transitive and type preserving subgroup. Let B ≤ G be an Iwahori subgroup of G and denoteby ( d s ) s ∈ S the thickness of X . Let q s = d − s , for s ∈ S . Then N q ( W ) ≅ L [ G, B ] .Proof. Denote by ϕ the isomorphism C q [ W ] → C [ G, B ] from Theorem 2.3. The Hecke von Neumannalgebra N q ( W ) is the completion of C q [ W ] in the GNS-representation associated with the tracialstate satisfying T ( q ) w ↦ δ w,e . This state is recovered as the composition of ϕ with the tracial vectorstate on C [ G, B ] associated with δ Be ∈ (cid:96) ( B / G ) . So we find that ϕ extends to a *-isomorphism L ( G, B ) ≅ N q ( W ) . 12actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski Proof of Corollary B.
Since λ G,B ( G ) ′ = R ( G, B ) ≅ L ( G, B ) , a combination of Theorem 3.4 and Theo-rem A applies to describe the structure of the quasi-regular representation. If ∑ w ∈ W q w is summable,then λ G,B = π ⊕ St for a semi-finite factor representation π and an irreducible representation St . If ∑ w ∈ W q w is not summable, then λ G,B is a semi-finite factor representation. We have to show thatneither π nor St (if it exists) can be finite representations. We observe that St is a subrepresentationof λ G,B and hence infinite dimensional. Let us now show that π is of type II ∞ . For a contradiction, letus assume that it is of type II , hence finite. By [BH19a, Theorem 15.D.5] the quotient G / ker π has acompact open normal subgroup, say L . Its preimage N ⊴ G must satisfy N B = G by [Tit64, Proposi-tion 2.5] (see alternatively [AB08, Lemma 6.61]). So N has finite index in G , since G / N ≅ B /( B ∩ N ) and B ∩ N is open inside the compact group B . So also L = N / ker π has finite index inside G / ker π which implies that the latter group is compact. We reach a contradiction, since compact groups haveno type II representations.Let us now show that if ∑ w ∈ W d − w < ∞ , then the irreducible representation St does not admit anynon-zero K -invariant vector. We may assume that B ≤ K and pick s ∈ S such that BsB ⊂ K . Indeed,if B is the stabiliser of C , we can pick K to be the stabiliser of a vertex adjacent to the facet labelledby s . Denote by p the minimal projection in L ( G, B ) , which exists according to Theorem A. It sufficesto show that p K p = . Combining the isomorphism of Hecke algebras Theorem 2.3 with the calculationof eigenvalues for T ( q ) s in Proposition 2.2, we find that BsB p = − p .On the other hand BsB p K = p K follows from the definition of the convolution product on C c ( G ) ,since BsB ⊂ K . This shows that p K p = must hold. Example 3.5.
The statement of Corollary B leaves open the possibility that for a given G an irreducibleIwahori-spherical square integrable representation exists. For large enough thickness of the underlyingbuilding, such a representation will always exist according to Corollary B. Indeed, if ( W, S ) is a Coxetergroup and X is the building of type ( W, S ) and thickness ( d s ) s ∈ S , then a closed, strongly transitiveand type preserving subgroup of Aut ( X ) admits an irreducible Iwahori-spherical square integrablerepresentation if and only if ∑ w ∈ W q w < ∞ , where q s = d − s . This applies in particular to the full groupof type-preserving automorphisms Aut ( X ) + , which has finite index in Aut ( X ) .• We first discuss one example in which there always is an irreducible direct summand of λ G,B .Let W = Z / Z ⊕ ∗ Z / Z and S = { s, t, u } the set of its standard generators where u is chosen togenerate the free copy of Z / Z . Enumerating elements of W according their u -length, one findsthat ∑ w ∈ W q w = ( + q s + q t + q st ) + ( + q s + q t + q st ) ∑ n ∈ N ≥ ( q s + q t + q st ) n − q nu .It follows that ∑ w ∈ W q w < ∞ if and only if q u (( + q s )( + q t ) − ) < . The last inequality isequivalent to ( + d s )( + d t ) d s d t < + d u , which holds for all allowable thicknesses, since d u ≥ .• Consider then the example W = Z / Z ⊕ ∗ Z / Z ⊕ with generators S = { s, t, u, v } chosen such thatthe pairs s, t and u, v commute. Reasoning as before, we calculate ∑ w ∈ W q w = ( + q s + q t + q st ) + ( + q s + q t + q st ) ∑ n ∈ N ≥ ( q s + q t + q st ) n − ( q u + q v + q uv ) n .Hence ∑ w ∈ W q w < ∞ if and only if ( ( + d s )( + d t ) d s d t − ) ( ( + d u )( + d v ) d u d v − ) < .This inequality is satisfied for all thick buildings of type ( W, S ) , since d s , d t , d u , d v ≥ must hold.13actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski • We finally also find examples where no irreducible Iwahori-spherical discrete series representationexists. Taking W = Z / Z ∗ k for k ≥ with its set of Coxeter generators S there are k ( k − ) n − words of length n ≥ in W . Hence, for a building of type ( W, S ) and uniform thickness d , wefind ∑ w ∈ W q w = + ∑ n ∈ N ≥ k ( k − ) n − d n ,which is summable if and only if d > k − . In this section we will develop a systematic point of view on the isomorphism C [ W ] ≅ C q [ W ] consideredin [Dav08, p.358, Note 19.2] and [CKL19, Proposition 4.2]. We will consider a special class of whatshould be thought of as graph products of representations. The construction of such representationsfollows a standard pattern known from free products of Hilbert spaces [Avi82; Voi85], which in turn wasthe basis of the introduction of graph product of Hilbert spaces as considered in [CF17, Section 2.1].We will briefly describe the graph product of representations. Using the notation and terminologyof [CF17], let Γ be a simplicial graph, let G v , v ∈ V Γ be a family of groups index by the vertices of Γ and denote by ∏ v, Γ G v the graph product, obtained as the quotient of the free product G Γ = ∗ v ∈ Γ G v by the commutation relations [ g, h ] = e for g ∈ G v , h ∈ G w and v ∼ Γ w . Suppose that for every v ∈ V Γ we are given a unitary representation π v ∶ G v → B ( H v ) and a unit vector ξ v ∈ H v . Let H denote thegraph product of Hilbert spaces associated to the family ( H v , ξ v ) v ∈ Γ . It is then easy to see that usingthe universal property of G Γ and injective unital *-homomorphisms λ v ∶ B ( H v ) → B ( H ) described in[CF17, Section 2] we obtain a representation π Γ ∶ G Γ → B ( H ) such that π Γ ∣ G v is unitarily equivalentto a multiple of the representation π v . The resulting representation does depend on the choice of thevectors ξ v , as noted already in [Avi82].Given a right-angled Coxeter system ( W, S ) and s ∈ S , the s-length of words provides a bijection W ≅ W /⟨ s ⟩ × ⟨ s ⟩ . Applying this bijection to the natural basis of (cid:96) ( W ) , we obtain a unitary U s ∶ (cid:96) ( W ) → (cid:96) ( W /⟨ s ⟩) ⊗ C . Proposition 4.1.
Let ( W, S ) be a right-angled Coxeter system and π s ∶ Z / Z → U ( ) , s ∈ S a familyof unitary representations. The formula π ( s ) = ( U s ) − ( id (cid:96) ( W /⟨ s ⟩) ⊗ π s ( )) U s for s ∈ S defines a unitary representation π ∶ W → U ( (cid:96) ( W )) .Proof. From the definition it is clear that π defines a unitary representation ˜ π of the free product group ˜ W = ∗ S Z / Z . We have to show that it descends to a representation of W by verifying the commutationrelation [ ˜ π ( s ) , ˜ π ( t )] = for all s, t ∈ S that commute. So fix different commuting elements s, t ∈ S .Since ⟨ s, t ⟩ = ⟨ s ⟩ × ⟨ t ⟩ ⊂ W , the s and t -length on W give rise to a bijection W ≅ W /⟨ s, t ⟩ × Z / Z × Z / Z ,which defines a unitary U ∶ (cid:96) ( W ) → (cid:96) ( W /⟨ s, t ⟩) ⊗ C ⊗ C . We have ˜ π ( s ) = U − ( id ⊗ π s ( ) ⊗ id ) U ˜ π ( t ) = U − ( id ⊗ id ⊗ π t ( )) U .So indeed [ ˜ π ( s ) , ˜ π ( t )] = , showing that π is well-defined.If we wish to put the construction above explicitly in the framework of graph products of repre-sentations mentioned before the lemma, we should only note the natural identifications, exploited forexample in [Cas20], of W with the graph product of order 2 groups, and of (cid:96) ( W ) with the graphproduct Hilbert space H Γ (where each H s for s ∈ V Γ = S is the two-dimensional vector space (cid:96) (⟨ s ⟩) with the chosen unit vector δ e ) and verify that the above construction is identical with the graphproduct representation. 14actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski Example 4.2.
Let us understand graph products of two-dimensional unitary representations arising inthe study of right-angled Coxeter groups in more detail. The spectral theorem provides the decompo-sition of representation spaces
Rep ( Z / Z , (cid:96) ( Z / Z )) = Rep ( Z / Z , C ) = { } ⊔ {− } ⊔ S .The two isolated components arise from the trivial representation and the sign representation. Thecomponent S ≅ U ( )/ T arises from a direct sum decomposition of C into a pair of orthogonalsubspaces on which Z / Z acts trivially and by the sign representation, respectively. An explicit de-scription of this component is obtained after observing that the nontrivial element ∈ Z / Z must actby a self-adjoint trace-free matrix ( a − a ) + ( zz ) ,for which the values a ∈ R and z ∈ C must satisfy a + ∣ z ∣ = .Applying Proposition 4.1 to a right-angled Coxeter system ( W, S ) , we obtain for each multiparameter ( a , z ) = ( a s , z s ) s ∈ S ∈ ( S ) S a unitary representation ˜ λ a , z of W on (cid:96) ( W ) . Denoting by σ ( s ) δ w = (− ) ∣ w ∣ s δ w = ⎧⎪⎪⎨⎪⎪⎩ δ w if ∣ sw ∣ > ∣ w ∣− δ w if ∣ sw ∣ < ∣ w ∣ .the alternating sign representation of W on (cid:96) ( W ) , we have ˜ λ a , z ( s ) = a s σ ( s ) + z s λ ( s ) , if z ∈ R S . Inparticular, if ( a , z ) = ( , ) s ∈ S , then ˜ λ a , z = λ is the left-regular representation of W .We observe that given ( a , z ) ∈ ( S ∖ {( , ) , (− , )}) S we can put ϕ s = z s ∣ z s ∣ ϕ w = ϕ s ⋯ ϕ s n for a reduced expression w = s ⋯ s n and obtain a unitary equivalence ˜ λ a , z ≅ ˜ λ a , ∣ z ∣ implemented by the unitary U ∶ (cid:96) ( W ) → (cid:96) ( W ) whichsatisfies U δ w = ϕ w δ w . We will thus write in what follows ˜ λ a = ˜ λ a , z for a ∈ (− , ) S putting z s = √ − a s for all s ∈ S .The identification of Hecke operator algebras of right-angled Coxeter groups with operator algebrasgenerated by their unitary representations was already observed in [Dav08, p.358, Note 19.2] andin [CKL19, Proposition 4.2]. These proofs admit a convenient reformulation by means of the graphproduct, which we provide for illustration. We formulate it for the Hilbert space case, but it could bealso stated in the (cid:96) r -framework when taking into account bounded, non-isometric representations on (cid:96) r ( W ) . Proposition 4.3.
Let ( W, S ) be a right-angled Coxeter system and a ∈ (− , ) S . Define q ∈ ( , ] S by q / s = − ∣ a s ∣ z s .Then ˜ λ a ( W ) ′′ ≅ N q ( W ) .Proof. For (cid:15) ∈ {− , } S given by (cid:15) s = sign ( a s ) , we put (cid:15) z = ( (cid:15) s z s ) s ∈ S . Then (cid:15) s ˜ λ a ( s ) = ˜ λ ∣ a ∣ ,(cid:15) z ( s ) for all s ∈ S and ˜ λ ∣ a ∣ ,(cid:15) z ≅ ˜ λ ∣ a ∣ . This allows us to reduce the considerations to the case where a s ≥ for all s ∈ S .Since both the Iwahori-Hecke algebras and the representations ˜ λ a arise as graph products, it suffices15actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski to consider the case W = Z / Z and match entries of × -matrices. The ansatz ˜ λ a ( s ) = c + dT ( q ) s forreal parameters c, d ∈ R yields the equality ( a s z s z s − a s ) = ⎛⎝ c dd c + d q s − q / s ⎞⎠ .Employing the identity z s = √ − a s , a short calculation yields q / s = − a s z s , finishing the proof of theproposition.We would like to note here that the above result points the way to constructing and studying interest-ing deformations of group algebras associated say to RAAGs via the graph products of representations,using the graph product structure of the groups in question and replacing the left regular representa-tion of the individual components by their ‘perturbations’, as was done above with the representationsof order 2 groups.By Theorem A, we know that ˜ λ a is a direct sum of a finite factor representation and a representationwhose dimension is at most one. Let us first identify the additional direct summand, slightly refiningthe picture of Hecke eigenvectors as considered in Proposition 2.2 and allowing us to pick up the signsof a parameter a ∈ (− , ) S . Instead of using direct calculations, it can be deduced from the descriptionof Hecke eigenvalues in [Dav08, Corollary 19.2.9] (and earlier left-sided computations in [Dav08]). Proposition 4.4.
Let ( W, S ) be a right-angled Coxeter group, a ∈ (− , ) S and set for each s ∈ Sq s = sign ( a s ) − ∣ a s ∣ z s = ⎧⎪⎪⎨⎪⎪⎩ − a s z s if a s ≥ − + a s z s if a s < .Assume that ∑ w ∈ W q w is absolutely summable and put ξ = ∑ w ∈ W q w δ w ∈ (cid:96) ( W ) . Then ˜ λ a ( s ) ξ = sign ( a s ) ξ .Proof. We write out the action of s on ξ as ˜ λ a ( s ) ξ = ( a s σ a ( s ) + z s λ ( s )) ξ = ∑ w ∈ W ∣ sw ∣>∣ w ∣ a s q w δ w + z s q w δ sw + ∑ w ∈ W ∣ sw ∣<∣ w ∣ − a s q w δ w + z s q w δ sw .If ∣ sw ∣ > ∣ w ∣ , then ( ˜ λ a ( s ) ξ ) w = a s q w + z s q sw = q w ( a s + z s sign ( a s ) − ∣ a s ∣ z s ) = sign ( a s ) q w .If ∣ sw ∣ < ∣ w ∣ we can make use of the identity z s q − s = sign ( a s ) z s − ∣ a s ∣ = sign ( a s ) − a s − ∣ a s ∣ = a s + sign ( a s ) and find that ( ˜ λ a ( s ) ξ ) w = − a s q w + z s q sw = q w (− a s + z s q − s ) = sign ( a s ) q w . Notation 4.5.
Let ( W, S ) be an irreducible, right-angled Coxeter system with at least three generatorsand let a ∈ (− , ) S . We write σ a ∶ W → {− , + } for the character satisfying σ a ( s ) = sign ( a s ) , s ∈ S .16actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski Further, we write λ a = ⎧⎪⎪⎨⎪⎪⎩ ˜ λ a if σ a /≤ λ a ˜ λ a ⊖ σ a if σ a ≤ λ a By Theorem A, Proposition 4.3 and Proposition 4.4, all λ a are finite factor representations of W .Let us now state some lemmas preparing for the proof of Theorem C. Lemma 4.6.
Let ( W, S ) be a right-angled Coxeter system, let a ∈ (− , ) S and let τ a be the characterof ( ˜ λ a , δ e ) . • For s ∈ S , we have τ a ( s ) = a s . • For s, t ∈ S different letters, we have τ a ( st ) = a s a t . • For s, t, u ∈ S pairwise different letters, we have τ a ( stu ) = a s a t a u . • If s, t ∈ S do not commute, then τ a ( stst ) = a s + a t − a s a t .Proof. We note that if w = s ⋯ s n is a reduced word in W , then τ a ( w ) is a sum over subwords of w which are trivial. Its coefficients are determined as a products of ± a s and z s , where s runs over theletters in the subword. Writing these expressions down explicitly, yields the statement of the lemma.In particular, τ a ( s ) = a s and τ a ( st ) = a s a t and τ a ( stu ) = a s a t a u are clear. The last statement followsfrom the calculation τ a ( stst ) = a t a s + a t z s + z t a s = a t a s + a t ( − a s ) + ( − a t ) a s = a s + a t − a s a t . Lemma 4.7.
Let ( W, S ) be an irreducible, right-angled Coxeter group and let a , b ∈ (− , ) S . Denoteby τ a , τ b the characters of ( ˜ λ a , δ e ) and ( ˜ λ b , δ e ) , respectively. Assume that there are c, d ∈ R such that τ a = cσ a + dσ b + ( − c − d ) τ b .If there is s ∈ S such that a s = b s , then a = b .Proof. The statement is trivial if S contains only one element. Without loss of generality, we mayassume that a s ≥ , which simplifies notation. We apply Lemma 4.6 to obtain a s = c + d + ( − c − d ) b s = c + d + ( − c − d ) a s .Simplifying this equality, we obtain a s ( c + d ) = c + d , which implies c + d = , since a s ≠ . Let now t ∈ S ∖ { s } be arbitrary. If sign ( a t ) = sign ( b t ) , Lemma 4.6 yields a t = sign ( a t )( c + d ) + ( − c − d ) b t = b t .If sign ( a t ) ≠ sign ( b t ) , then the identity c + d = implies a t = sign ( a t )( c − d ) + ( − c − d ) b t = sign ( a t ) c + b t .Similarly, a s a t = sign ( a t ) c + a s b t .Taking the difference of these identities gives us ( − a s ) a t = ( − a s ) b t ,which allows to conclude a t = b t , because a s ≠ . Since t ∈ S ∖ { s } was arbitrary, this concludes theproof of the lemma. 17actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski Proof of Theorem C.
Let us assume that λ a ≅ λ b for two parameters a , b ∈ (− , ) S . Denote by τ a , τ b the characters of ( ˜ λ a , δ e ) and ( ˜ λ b , δ e ) , respectively. Then Proposition 4.4 implies that there are scalars c, d ∈ R such that c + d ∈ [ , ) and τ a = cσ a + dσ b + ( − c − d ) τ b .Replacing a and b by (∣ a s ∣) s ∈ S and ( sign ( a s ) b s ) s ∈ S , respectively, we may assume that a s ≥ for all s ∈ S . We distinguish three cases. Case 1 . There are two letters s, t ∈ S such that b s , b t ≥ .Putting γ = c + d , Lemma 4.6 implies that a s = γ + ( − γ ) b s a t = γ + ( − γ ) b t a s a t = γ + ( − γ ) b s b t .Substituting the first two expressions into the third, and simplifying, we obtain = ( γ − γ ) + γ ( − γ )( b t + b s ) + (( − γ ) − ( − γ )) b s b t = γ ( γ − )( b s − )( b t − ) .Since γ, b s , b t ≠ , we find that γ = and thus a s = b s . So Lemma 4.7 shows that a = b . Case 2 . There is exactly one u ∈ S such that b u ≥ .We will obtain a contradiction in this case. Take s ∈ S ∖ { u } . Then Lemma 4.6 provides the identities a s = c − d + ( − c − d ) b s a u a s = c − d + ( − c − d ) b u b s ,which we subtract in order to obtain ≤ a s ( − a u ) = ( − c − d ) b s ( − b u ) .Since b s < and − b u > , this implies − c − d ≤ . The latter is a contradiction to c + d ∈ [ , ) . Case 3 . We have b s < for all s ∈ S .Again we aim for a contradiction. Take three pairwise different letters s, t, u ∈ S . From Lemma 4.6, weobtain that a s = c − d + ( − c − d ) b s a t = c − d + ( − c − d ) b t a u a s = c + d + ( − c − d ) b u b s a u a t = c + d + ( − c − d ) b u b t .We subtract the second identity from the first one and the fourth from the third one, and we obtain a s − a t = ( − c − d )( b s − b t ) a u ( a s − a t ) = ( − c − d ) b u ( b s − b t ) .Since a u ≠ b u , this implies a s = a t . Because c + d ≠ , we also obtain the identity b s = b t . Similarly a s = a u .Let us now take pairwise different letters s, t, u ∈ S . Evaluating τ a = cσ a + dσ b + ( − c − d ) τ b on thereduced words s and stu , we obtain a s = c − d + ( − c − d ) b b a s = a s a t a u = c − d + ( − c − d ) b s b t b u = c − d + ( − c − d ) b s .18actorial Hecke von Neumann algebras by Sven Raum and Adam Skalski Subtracting the second from the first identity we obtain a s − a s = ( − c − d )( b s − b s ) .Since a s ∈ [ , ) and b s ∈ (− , ) , this implies that − c − d < , which is a contradiction to c + d ∈ [ , ) . Proof of Corollary D.
Denote by χ a the character of λ a , a ∈ (− , ) S . We first show that the assignment a ↦ χ a is continuous. Denoting by τ a the character of ˜ λ a , it is clear that a ↦ τ a is continuous. Write F ( a ) = ∑ w ∈ W q w ∈ R > ∪ {∞} , where q s = ( − ∣ a s ∣ z s ) = ( − ∣ a s ∣) − a s = − ∣ a s ∣ + ∣ a s ∣ .Thus F is a continuous function with values in R > ∪ {∞} and hence also / F , which takes values in R ≥ , is continuous. If F ( a ) < ∞ , then ξ a = ∑ w ∈ W q w δ w ∈ (cid:96) ( W ) . The projection p on C ξ a is central in N q ( W ) = ˜ λ a ( W ) ′′ by Theorem A and satisfies ⟨ pδ e , δ e ⟩ = ∥ ξ a ∥ ∣⟨ ξ a , δ e ⟩∣ = F ( a ) .Hence, the definition of λ a in Notation 4.5 shows that χ a = − F ( a ) ( τ a − F ( a ) σ a ) ,showing that χ a indeed continuously depends on a ∈ (− , ) S . Since χ is the regular character, itremains to observe that the map a ↦ χ a is injective by Theorem C and thus a homeomorphism ontothe image of a neighbourhood of ∈ R ∣ S ∣ . References [AB08] P. Abramenko and K. S. Brown.
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Sven RaumDepartment of MathematicsStockholm UniversitySE-106 91 StockholmSwedenandInstitute of Mathematics of thePolish Academy of Sciencesul. Śniadeckich 800-656 WarszawaPoland [email protected]
Adam SkalskiInstitute of Mathematics of thePolish Academy of Sciencesul. Śniadeckich 800-656 WarszawaPoland [email protected]@impan.pl