The Weighted Atiyah Conjecture for Hecke–von Neumann algebras of Coxeter groups beyond the infinite dihedral group

The Weighted Atiyah Conjecture of Okun and Scott predicts that, for a Coxeter group $W$ with Hecke algebra $\mathbb R_{\mathbf q}W$ and positive real multiparameter $\mathbf q$, the von Neumann dimension of the kernel of right multiplication by any matrix over the algebra generated by the spherical idempotents $a_J$ is a finite sum $\sum_{J}n_J(\mathbf q)/W_J(\mathbf q)$, where $W_J(\mathbf q)$ is the growth polynomial of the finite parabolic subgroup $W_J$ and the $n_J$ are piecewise-constant integer-valued functions of $\mathbf q$; in the right-angled case the same conclusion is conjectured for matrices over the integral group ring $\mathbb ZW$. This was proved in 2014 only for the infinite dihedral group. The problem is to prove or refute the conjecture for right-angled Coxeter groups beyond the infinite dihedral group — already open for the free product $(\mathbb Z/2)^{*3}$ — and, at full generality, for arbitrary Coxeter groups.

Background

The Atiyah conjecture concerns the $L^2$-theory of discrete groups. For a group $G$, the Hilbert space $L^2G$ is the completion of the group algebra $\mathbb RG$ in the inner product making the group elements an orthonormal basis, and the group von Neumann algebra $\mathcal NG$ is the algebra of bounded operators on $L^2G$ commuting with the right $G$-action. Any closed, left-$G$-invariant subspace $V\subseteq(L^2G)^n$ has a von Neumann dimension $\dim_GV$, obtained by applying the canonical trace of $\mathcal NG$ to the orthogonal projection onto $V$; it takes values in $[0,n]$ and agrees with ordinary dimension divided by the group order in the finite case. Kernels of right-multiplication operators $R_M$ by matrices $M$ over the rational group ring compute the $L^2$-Betti numbers of spaces with a $G$-action. The Atiyah conjecture predicts that $\dim_G\ker R_M$ is always rational, and the Strong Atiyah conjecture predicts it lies in the additive subgroup $\Lambda\subseteq\mathbb R$ generated by the reciprocals $1/|H|$ of the orders of the finite subgroups $H\le G$. In full generality the conjecture is false — counterexamples with irrational (even transcendental) kernel dimensions are due to Austin, Grabowski, and Pichot–Schick–Żuk — but all known counterexamples involve groups with finite subgroups of unbounded order, and the strong form remains open (and widely believed) for groups with bounded torsion.

A Coxeter group $W$ is given by generators $S$ and relations $s^2=1$ and $(st)^{m_{st}}=1$ for integers $m_{st}\ge2$; it is right-angled when every $m_{st}$ is either 2 (so $s$ and $t$ commute) or $\infty$ (no relation between $s$ and $t$). A subset $J\subseteq S$ is spherical when the parabolic subgroup $W_J=\langle J\rangle$ is finite; $\mathcal S$ denotes the collection of spherical subsets. The Hecke algebra $\mathbb R_{\mathbf q}W$ with real multiparameter $\mathbf q=(q_s)_{s\in S}$ (constant on conjugacy classes of generators) has an $\mathbb R$-basis $\{T_w\}_{w\in W}$ with multiplication $T_sT_w=T_{sw}$ if $|sw|>|w|$ and $T_sT_w=(q_s-1)T_w+q_sT_{sw}$ otherwise — a deformation of the group algebra, recovered at $\mathbf q=\mathbf 1$. Writing $q^w$ for the product of the parameters along a reduced expression for $w$, the algebra carries the inner product $\langle T_w,T_{w'}\rangle=q^w\delta_{w,w'}$; its completion is the Hilbert space $L^2_{\mathbf q}W$, and the bounded operators commuting with the right Hecke action form the Hecke–von Neumann algebra $\mathcal N_qW$, which yields von Neumann dimensions $\dim_W^{\mathbf q}$ for closed Hecke-invariant subspaces. For each spherical $J$, the element $a_J=(\sum_{w\in W_J}T_w)/W_J(\mathbf q)$, where $W_J(\mathbf q)=\sum_{w\in W_J}q^w$ is the growth polynomial of $W_J$, is an idempotent whose right action is the orthogonal projection onto a summand of $\dim_W^{\mathbf q}$-dimension $1/W_J(\mathbf q)$.

For the reflection action of $W$ on its Davis complex (the standard contractible CW-complex on which $W$ acts properly and cocompactly by reflections), the weighted $L^2$-cochain complexes with respect to $\mathbf q$ decompose into these summands, and the resulting weighted $L^2$-Betti numbers govern the weighted $L^2$-cohomology. Davis–Dymara–Januszkiewicz–Okun proved that these Betti numbers are continuous functions of $\mathbf q$ and asked whether they are piecewise rational functions of $\mathbf q$. Okun–Scott reformulated and strengthened this question into an algebraic conjecture: for a matrix $M$ over the algebra $\mathcal AW$ generated over the field $\mathbb Q(\mathbf q)$ of rational functions in the parameters by the idempotents $a_J$, the von Neumann dimension of the kernel of right multiplication by the specialization $M_{\mathbf q}$ should be a sum $\sum_{J\in\mathcal S}n_J(\mathbf q)/W_J(\mathbf q)$ with the $n_J$ piecewise-constant integer-valued functions of $\mathbf q$ (the Weighted Atiyah Conjecture, WAC). In the right-angled case, where a canonical isomorphism $\mathbb R_{\mathbf q}W\cong\mathbb RW$ identifies the $a_J$ with group-algebra idempotents, the same conclusion is conjectured for matrices over the integral group ring $\mathbb ZW$ (the Right-Angled Weighted Atiyah Conjecture, RAWAC).

Okun–Scott established both statements for the infinite dihedral group $W=D_\infty=\langle s,t\mid s^2=t^2=1\rangle$. The proof exploits the index-2 infinite cyclic subgroup $G=\langle st\rangle$: after passing to the commutative Laurent polynomial ring $\mathbb RG$, the key technical fact is that the action of $st$ on $L^2_{\mathbf q}G$ admits only the eigenvalues $+1$ and $-1$, which yields the fine orthogonal decomposition $L^2_{\mathbf q}W=K_+\oplus K_-\oplus K_\emptyset\oplus K_\emptyset s$ and reduces every kernel to a trivial-or-full map on each summand; the resulting dimension is $n_\emptyset+n_s/(1+q_s)+n_t/(1+q_t)$, piecewise constant with jumps only along the curves $q_s=q_t$ and $q_sq_t=1$. The summand $K_-$ is a new invariant subspace absent from both the ordinary group-algebra picture and the general decomposition of Davis et al. By contrast, in the undeformed case $\mathbf q=\mathbf 1$ the Strong Atiyah Conjecture is known for right-angled Artin and Coxeter groups (Linnell–Okun–Schick, 2012) and, as of 2025, for all Coxeter groups (Fisher–Hughes, appendix to Hughes–Möller–Varghese) — but those subgroup-ring and division-closure techniques have no analogue for Hecke algebras, where outside the right-angled case subgroups of $W$ do not even correspond to subalgebras of $\mathbb R_{\mathbf q}W$.

Problem Statement

Let $(W,S)$ be a Coxeter system, $\mathcal S$ the set of spherical subsets of $S$ (those $J\subseteq S$ whose parabolic subgroup $W_J=\langle J\rangle$ is finite), $W_J(\mathbf q)=\sum_{w\in W_J}q^w$ the growth polynomial of a finite parabolic subgroup, where $q^w$ denotes the product of the parameters $q_s$ along a reduced expression for $w$, and $L^2_{\mathbf q}W$ the Hilbert space completion of the Hecke algebra $\mathbb R_{\mathbf q}W$ with $\langle T_w,T_{w'}\rangle=q^w\delta_{w,w'}$, on which the Hecke–von Neumann algebra $\mathcal N_qW$ acts and yields the dimension function $\dim_W^{\mathbf q}$. The Right-Angled Weighted Atiyah Conjecture (Okun–Scott) asserts: if $W$ is right-angled and $M$ is an $n\times m$ matrix with entries in the integral group ring $\mathbb ZW$, identified with a matrix over $\mathbb R_{\mathbf q}W$ via the canonical isomorphism, then

$$\dim_W^{\mathbf q}\ker R_M=\sum_{J\in\mathcal S}\frac{n_J(\mathbf q)}{W_J(\mathbf q)}$$

for all positive multiparameters $\mathbf q\in(\mathbb R_{>0})^S$ (constant on conjugacy classes of generators), where $R_M:(L^2_{\mathbf q}W)^n\to(L^2_{\mathbf q}W)^m$ is right multiplication by $M$ and the $n_J$ are piecewise-constant, integer-valued functions of $\mathbf q$. The Weighted Atiyah Conjecture asserts the same for arbitrary Coxeter groups, with $M$ a matrix over the algebra $\mathcal AW$ generated over $\mathbb Q(\mathbf q)$ by the idempotents $a_J=(\sum_{w\in W_J}T_w)/W_J(\mathbf q)$ and $R_M$ the right multiplication by its specialization $M_{\mathbf q}$.

Okun–Scott proved both statements when $W$ is the infinite dihedral group. The problem is to determine whether they hold beyond that case:

(1) Does the Right-Angled Weighted Atiyah Conjecture hold for every right-angled Coxeter group — already open for the free product $(\mathbb Z/2)^{*3}$ of three copies of $\mathbb Z/2$ and, generally, for any right-angled Coxeter group that is neither finite nor the infinite dihedral group?

(2) Does the Weighted Atiyah Conjecture hold for arbitrary, not necessarily right-angled Coxeter groups?

A complete answer resolves both (1) and (2), and for each part it is either a proof for that part's class of groups or an explicit counterexample: a specific Coxeter group $W$ (right-angled for (1), arbitrary for (2)), a specific finite matrix $M$ over $\mathbb ZW$ for (1) (respectively over $\mathcal AW$ for (2)), and a specific parameter point $\mathbf q_0\in(\mathbb R_{>0})^S$ such that $\dim_W^{\mathbf q_0}\ker R_M$ is not expressible as $\sum_{J\in\mathcal S}n_J(\mathbf q)/W_J(\mathbf q)$ with integer piecewise-constant numerators. The closely related structural question identified in the source — whether analogues of the dihedral orthogonal decomposition $L^2_{\mathbf q}W=K_+\oplus K_-\oplus K_\emptyset\oplus K_\emptyset s$ and new invariant subspaces such as $K_-$ exist for other Coxeter groups — is the known methodological barrier, not an independent deliverable.

References

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