1 Introduction↩︎

The classical Ray–Singer analytic torsion [1] is a spectral invariant extracted from the de Rham complex of a closed manifold. The celebrated Cheeger–Müller theorem [2]–[5] asserts that this analytic torsion essentially coincides with the Reidemeister torsion, a topological invariant.

Rumin and Seshadri [6] have introduced an analytic torsion of the Rumin complex on contact manifolds [7]–[9] and showed that it coincides with the Ray–Singer torsion for 3-dimensional CR Seifert manifolds. Further computations for contact spheres and lens spaces have been carried out by Kitaoka [10], [11]. Recently, Albin and Quan [12] proved that the Rumin–Seshadri analytic torsion differs from the Ray–Singer torsion by the integral of a local quantity, which yet has to be identified explicitly.

An analytic torsion for the Rumin complex on more general filtered manifolds [13]–[16] has been proposed in [17]. This analytic torsion is only defined if the osculating algebras of the filtered manifold have pure cohomology. The latter assumption appears to be rather restrictive [17]. We are only aware of three types of filtered manifolds with this property: trivially filtered manifolds giving rise to the Ray–Singer torsion, contact manifolds giving rise the Rumin–Seshadri torsion, and generic rank two distributions in dimension five which are also known as (2,3,5) distributions [17].

A generic rank two distributions in dimension five is a rank two subbundle \(\mathcal{D}\) in the tangent bundle of a 5-manifold \(M\) such that Lie brackets of sections of \(\mathcal{D}\) span a subbundle \([\mathcal{D},\mathcal{D}]\) of rank three, and triple brackets of sections of \(\mathcal{D}\) span all of the tangent bundle, \([\mathcal{D},[\mathcal{D},\mathcal{D}]]=TM\). These geometric structures have first been studied by Cartan [18]. The Lie bracket of vector fields induces a fiberwise Lie bracket on the associated graded bundle, \[\mathfrak tM=\tfrac{TM}{[\mathcal{D},\mathcal{D}]}\oplus\tfrac{[\mathcal{D},\mathcal{D}]}{\mathcal{D}}\oplus\mathcal{D}.\] This is a locally trivial bundle of graded Lie algebras over \(M\) called the bundle of osculating algebras. Its fibers are all isomorphic to the 5-dimensional graded nilpotent Lie algebra \[\mathfrak g=\mathfrak g_{-3}\oplus\mathfrak g_{-2}\oplus\mathfrak g_{-1}\] with graded basis \(X_1,X_2\in\mathfrak g_{-1}\), \(X_3\in\mathfrak g_{-2}\), \(X_4,X_5\in\mathfrak g_{-3}\) and brackets \[\label{E:brackets} [X_1,X_2]=X_3,\quad[X_1,X_3]=X_4,\quad[X_2,X_3]=X_5.\tag{1}\]

The simply connected Lie group with Lie algebra \(\mathfrak g\) will be denoted by \(G\). The left invariant 2-plane field spanned by \(X_1\) and \(X_2\) provides a basic example of a (2,3,5) distribution on \(G\). Unlike contact or Engel structures, (2,3,5) distributions do have local geometry. A distribution of this type is locally diffeomorphic to the aforementioned left invariant distribution on \(G\) if and only if Cartan’s [18] harmonic curvature tensor, a section of \(S^4\mathcal{D}^*\), vanishes. This curvature tensor is constructed using an equivalent description of (2,3,5) distributions as regular normal parabolic geometries of type \((G_2,P)\) where \(G_2\) denotes the split real form of the exceptional Lie group and \(P\) denotes the parabolic subgroup corresponding to the longer root, see [18], [19] or [20]. Generic rank two distributions in dimension five have attracted quite some attention recently, cf. [17], [19], [21]–[37].

The Rumin complex associated to a (2,3,5) distribution on a 5-manifold \(M\) is a natural sequence of higher order differential operators \[\cdots\to\Gamma\bigl(\mathcal{H}^q(\mathfrak tM)\bigr)\xrightarrow{D_q}\Gamma\bigl(\mathcal{H}^{q+1}(\mathfrak tM)\bigr)\to\cdots\] where \(\mathcal{H}^q(\mathfrak tM)\) denotes the vector bundle obtained by passing to the fiberwise Lie algebra cohomology of \(\mathfrak tM\) with trivial coefficients. The Betti numbers are \(\mathop{\mathrm{rk}}\mathcal{H}^q(\mathfrak tM))=\dim H^q(\mathfrak g)=1,2,3,2,1\) for \(q=0,\dotsc,4\) and the Heisenberg order of the Rumin differential \(D_q\) is \(k_q=1,3,2,3,1\) for \(q=0,\dotsc,4\), see [38] and [39]. The Rumin differentials form a complex, \(D_{q+1}D_q=0\), that computes the de Rham cohomology of \(M\). Actually, there exist injective differential operators \(L_q\colon\Gamma(\mathcal{H}^q(\mathfrak tM))\to\Omega^q(M)\), embedding the Rumin complex as a subcomplex in the de Rham complex and inducing isomorphisms on cohomology. Twisting with a flat complex vector bundle \(F\) we obtain a complex of differential operators \[\label{E:Rumin46M} \cdots\to\Gamma\bigl(\mathcal{H}^q(\mathfrak tM)\otimes F\bigr)\xrightarrow{D_q}\Gamma\bigl(\mathcal{H}^{q+1}(\mathfrak tM)\otimes F\bigr)\to\cdots\tag{2}\] computing \(H^*(M;F)\), the de Rham cohomology of \(M\) with coefficients in \(F\). Rumin has shown that the sequence 2 becomes exact on the level of the Heisenberg principal symbol, see [13], [14], or [39]. Hence, the Rumin complex is a Rockland [40] complex, the analogue of an elliptic complex in the Heisenberg calculus, see [39] for more details.

A fiberwise graded Euclidean inner product \(g\) on \(\mathfrak tM\) and a fiberwise Hermitian inner product \(h\) on \(F\) give rise to \(L^2\) inner products on \(\Gamma(\mathcal{H}^q(\mathfrak tM)\otimes F)\) which in turn provide formal adjoints \(D_q^*\) of the Rumin differentials in 2 . Assuming \(M\) to be closed, the operator \(D_q^*D_q\) has an infinite dimensional kernel if \(q>0\) but the remaining part of its spectrum consists of isolated positive eigenvalues with finite multiplicities only. Moreover, \((D_q^*D_q)^{-s}\) is trace class for \(\Re s>10/2k_q\). The number ten appears here because this is the homogeneous dimension of the filtered manifold \(M\). Furthermore, the function \(\mathop{\mathrm{tr}}(D_q^*D_q)^{-s}\) admits an analytic continuation to a meromorphic function on the entire complex plane which is holomorphic at \(s=0\), see [17]. This permits to define the zeta regularized determinant \({\det}_*|D_q|\) by \[\label{E:edtDq46def} \log{\det}_*|D_q|:=-\tfrac12\tfrac\partial{\partial s}\big|_{s=0}\mathop{\mathrm{tr}}(D_q^*D_q)^{-s}.\tag{3}\] The notation \({\det}_*\) indicates that the zero eigenspace does not contribute, i.e., we are considering the regularized product of nonzero eigenvalues. Correspondingly, the complex powers are defined to vanish on the kernel of \(D_q^*D_q\). The analytic torsion \(\tau(M,\mathcal{D},F,g,h)\) is the graded determinant of the Rumin complex, i.e., \[\label{E:tau46def} \log\tau(M,\mathcal{D},F,g,h):=\sum_{q=0}^4(-1)^q\log{\det}_*|D_q|.\tag{4}\] By Poincaré duality, see [13], [14], or [17], we have \({\det}_*|D_0|={\det}_*|D_4|\) and \({\det}_*|D_1|={\det}_*|D_3|\), provided \(h\) is parallel. Hence, in this (unitary) case the torsion may be expressed in terms of three determinants, \[\tau(M,\mathcal{D},F,g,h)=\frac{{\det}_*^2|D_0|\cdot{\det}_*|D_2|}{{\det}_*^2|D_1|}.\]

Since \(\mathfrak g\) has pure cohomology, the grading automorphism \(\phi_\tau\in\mathop{\mathrm{Aut}}_\textrm{gr}(\mathfrak g)\) acts as a scalar on each cohomology, namely by \(\tau^{N_q}\) on \(H^q(\mathfrak g)\) where \(N_q=0,1,4,6,9,10\) for \(q=0,\dotsc,5\). These numbers are related to the Heisenberg orders of the Rumin differentials via the equation \(k_q=N_{q+1}-N_q\). We fix natural numbers \(a_q\) such that \(\kappa:=a_qk_q\) is independent of \(q\). The smallest possible choice would be \(a_q=6,2,3,2,6\) with \(\kappa=6\). In general, \(\kappa\) is a multiple of six. Then the Rumin–Seshadri operators \[\label{E:Delta46def} \Delta_q:=\bigl(D_{q-1}D_{q-1}^*\bigr)^{a_{q-1}}+\bigl(D_q^*D_q\bigr)^{a_q}\tag{5}\] are all of Heisenberg order \(2\kappa\). These operators are analytically much better behaved than \(D_q^*D_q\). Indeed, \(\Delta_q\) is a Rockland [40] operator and, thus, admits a parametrix in the Heisenberg calculus [39], [41], [42]. Moreover, \(\Delta_q^{-s}\) is trace class for \(\Re s>10/2\kappa\), and the zeta function \(\mathop{\mathrm{tr}}\Delta_q^{-s}\) admits an analytic continuation to a meromorphic function on the entire complex plane which is holomorphic at \(s=0\), see [43]. The analytic properties of the operator \(D_q^*D_q\) stated in the preceding paragraph can readily be deduced from the corresponding properties of \(\Delta_q\), see [17] for more details. Putting \[\label{E:zeta46M46def} \zeta_{M,\mathcal{D},F,g,h}(s):=\mathop{\mathrm{str}}(N\Delta^{-s}):=\sum_{q=0}^5(-1)^qN_q\mathop{\mathrm{tr}}(\Delta_q^{-s})\tag{6}\] the analytic torsion of the Rumin complex can be expressed in the form \[\label{E:tau46via46zeta} \tau(M,\mathcal{D},F,g,h)=\exp\left(\tfrac1{2\kappa}\zeta'_{M,\mathcal{D},F,g,h}(0)\right)\tag{7}\] which is analogous to the formulas for the Ray–Singer torsion in [1] and the Rumin–Seshadri torsion in [6], see [17] for more details.

It turns out to be convenient to incorporate the zero eigenspaces of \(\Delta_q\) and consider the analytic torsion of the Rumin complex as a norm \(\|-\|^{\mathop{\mathrm{sdet}}H^*(M;F)}_{\mathcal{D},g,h}\) on the graded determinant line \[\mathop{\mathrm{sdet}}H^*(M;F)=\bigotimes_{q=0}^5\bigl(\det H^q(M;F)\bigr)^{(-1)^q}.\] Basic properties of this torsion have been established in [17]. In the acyclic case, it reduces to (the reciprocal of) \(\tau(M,\mathcal{D},F,g,h)\). The Ray–Singer torsion, too, is best regarded as a norm on this graded determinant line [2], [44]. We will denote it by \(\|-\|_\textrm{RS}^{\mathop{\mathrm{sdet}}H^*(M;F)}\). The Ray–Singer torsion does not depend on metric choices since the dimension of \(M\) is odd.

In this paper we will determine the analytic torsion of the Rumin complex on nilmanifolds \(\Gamma\setminus G\) where \(\Gamma\) is a lattice in \(G\), i.e., a cocompact discrete subgroup in \(G\). We consider any left invariant (2,3,5) distribution \(\mathcal{D}_G\) on \(G\) and any left invariant fiberwise graded Euclidean inner product on \(\mathfrak tG\). These descend to a (2,3,5) distribution \(\mathcal{D}_{\Gamma\setminus G}\) on the nilmanifold \(\Gamma\setminus G\) and a fiberwise graded Euclidean inner product \(g_{\Gamma\setminus G}\) on \(\mathfrak t(\Gamma\setminus G)\). For a unitary representation \(\rho\colon\Gamma\to U(k)\) we let \(h_\rho\) denote the canonical (parallel) fiberwise Hermitian inner product on the associated flat complex vector bundle \(F_\rho:=G\times_\rho\mathbb{C}^k\) over \(\Gamma\setminus G\).

We are now in a position to state our main result.

In the situation of this theorem, the Ray–Singer torsion is known to be trivial too, \(\tau_\textrm{RS}(\Gamma\setminus G;F_\chi)=1\), see Lemma 11 below. Hence, it coincides with the analytic torsion of the Rumin complex. More generally, we have:

These are the first (2,3,5) distributions for which the analytic torsion of the Rumin complex has been computed. In [17] a partial result of this type has been obtained by exploiting a large discrete symmetry group of the distribution \(\mathcal{D}_{\Gamma\setminus G}\).

The proof given below is based on the decomposition of the Rumin complex over \(\Gamma\setminus G\) into a countable direct sum \(D_*=\bigoplus_\rho m(\rho)\cdot\rho(D_*)\) of Rumin complexes in irreducible unitary representations \(\rho\) of \(G\), denoted by \(\rho(D_*)\). The multiplicities \(m(\rho)\) of the contributing representations are known explicitly through a formula due to Howe [45] and Richardson [46] and involve counting the number of solutions to certain quadratic congruences, cf. Lemma 5(III), 38 , or 54 below. The zeta function in 6 decomposes accordingly, \[\label{E:zzz} \zeta_{\Gamma\setminus G,\mathcal{D}_{\Gamma\setminus G},F_\chi,g_{\Gamma\setminus G},h_\chi}(s)=\sum_\rho m(\rho)\cdot\zeta_\rho(s)\tag{8}\] where \(\zeta_\rho(s)\) denotes the zeta function associated with \(\rho(D_*)\). The values \(\zeta_\rho(0)\) and \(\zeta'_\rho(0)\) have been determined in [30] for every irreducible unitary representation \(\rho\). Even though the sum on the right hand side in 8 converges only for \(\Re s>10/2\kappa\), we are able to conclude \(\zeta_{\Gamma\setminus G,\mathcal{D}_{\Gamma\setminus G},F_\chi,g_{\Gamma\setminus G},h_\chi}'(0)=0\) via regularization, whence the theorem stated above. This is the subtlest part of the paper at hand and builds on further properties of \(\zeta_\rho(s)\) obtained in [30]. Decomposing \[\sum_\rho m(\rho)\cdot\zeta_\rho(s)=\zeta_{\mathrm{I},\Gamma,\chi}(s)+\zeta_{\mathrm{I\!I},\Gamma,\chi}(s)+\zeta_{\mathrm{I\!I\!I},\Gamma,\chi}(s)\] according to the three types of irreducible unitary representations of \(G\), we will use Epstein zeta functions to obtain the analytic continuation of each of the three summands, cf. 57 , 62 , and 77 below.

Alternatively, one could try to extend Albin and Quan’s [12] analysis of the sub-Riemannian limit in order to show that the torsion of the Rumin complex of a (2,3,5) distribution differs from the Ray–Singer torsion by the integral of a local quantity cf. [12]. This, too, would immediately imply the corollary stated above.

The remaining part of this paper is organized as follows. In Section 2 we provide an explicit description of all lattices in \(G\). In Section 3 we use the aforementioned result due to Howe [45] and Richardson [46] to decompose the space of sections of \(F_\chi\) into irreducible \(G\)-representations. In Section 4 we describe the corresponding decomposition of the zeta function \(\zeta_{\Gamma\setminus G,\mathcal{D}_{\Gamma\setminus G},F_\chi,g_{\Gamma\setminus G},h_\chi}(s)\). In Section 5, building on results obtained in [30], we evaluate the derivative of this zeta function at \(s=0\). In Section 6 we derive the theorem and corollary stated above.

2 Lattices↩︎

In this section we provide explicit descriptions of all lattices in \(G\).

Let \(X_1,\dotsc,X_5\) be a graded basis of \(\mathfrak g\) with brackets as in 1 . For the sake of notational simplicity we will use this basis to identify \(\mathfrak g\) with \(\mathbb{R}^5\), that is, a vector \((x_1,\dotsc,x_5)^t\in\mathbb{R}^5\) will be identified with \(\sum_{i=1}^5x_iX_i\in\mathfrak g\).

The exponential map provides a diffeomorphism \(\exp\colon\mathfrak g\to G\). Using the Baker–Campbell–Hausdorff formula we find \[\exp\begin{pmatrix}x_1\\\vdots\\x_5\end{pmatrix} \exp\begin{pmatrix}y_1\\\vdots\\y_5\end{pmatrix} =\exp\begin{pmatrix}z_1\\\vdots\\z_5\end{pmatrix}\] where \[\label{E:zxy} z=x\cdot y :=\left(\begin{array}{l} x_1+y_1\\ x_2+y_2\\ x_3+y_3+\frac{x_1y_2-x_2y_1}{2}\\x_4+y_4+\frac{x_1y_3-x_3y_1}{2}+\frac{(x_1-y_1)(x_1y_2-x_2y_1)}{12}\\ x_5+y_5+\frac{x_2y_3-x_3y_2}{2}+\frac{(x_2-y_2)(x_1y_2-x_2y_1)}{12} \end{array}\right).\tag{9}\]

The center of \(G\) will be denoted by \(Z\). Clearly, \(Z=\exp(\mathfrak z)\) where \(\mathfrak z\) denotes the center of \(\mathfrak g\) which is spanned by \(X_4,X_5\). For the group of commutators we have \([G,G]=\exp([\mathfrak g,\mathfrak g])\) and the derived subalgebra \([\mathfrak g,\mathfrak g]\) is spanned by \(X_3,X_4,X_5\).

For \(r\in\mathbb{N}\) and \(e,f,g,h,u,v\in\mathbb{Q}\) we consider \(\tilde{\gamma}_i\in\mathfrak g\) defined by \[\label{E:Gamma46tgen} \tilde{\gamma}_1=\begin{pmatrix}1\\0\\0\\0\\0\end{pmatrix},\quad \tilde{\gamma}_2=\begin{pmatrix}0\\1\\0\\0\\0\end{pmatrix},\quad \tilde{\gamma}_3=\begin{pmatrix}0\\0\\1/r\\u/2r\\v/2r\end{pmatrix},\quad \tilde{\gamma}_4=\begin{pmatrix}0\\0\\0\\e\\f\end{pmatrix},\quad \tilde{\gamma}_5=\begin{pmatrix}0\\0\\0\\g\\h\end{pmatrix},\tag{10}\] and let \(\Gamma\) denote the subgroup of \(G\) generated by the exponentials \[\gamma_i:=\exp\tilde{\gamma}_i,\qquad i=1,\dotsc,5.\] Let \(\Gamma''\) denote the lattice in \(\mathbb{R}^2\) generated by \[\label{E:Gamma3939} \begin{pmatrix}1/r\\0\end{pmatrix},\quad \begin{pmatrix}0\\1/r\end{pmatrix},\quad \begin{pmatrix}\frac{u-1}{2}\\\frac{v-1}{2}\end{pmatrix},\quad \begin{pmatrix}e\\f\end{pmatrix},\quad \begin{pmatrix}g\\h\end{pmatrix}.\tag{11}\]

The following generalizes [17].

The natural homomorphism \(p\colon G\to G/[G,G]\cong\mathbb{R}^2\) gives rise to a short exact sequence of abelian groups \[\label{E:abelGamma} 0\to\frac{\Gamma\cap[G,G]}{[\Gamma,\Gamma]}\to\frac{\Gamma{[\Gamma,\Gamma]}}{\to} p(\Gamma)\to0\tag{20}\] where \(\frac{\Gamma\cap[G,G]}{[\Gamma,\Gamma]}\) is a finite abelian group and \(p(\Gamma)\cong\mathbb{Z}^2\). In particular, the sequence splits and we obtain an isomorphism \[\label{E:Gamma46ab} \frac{\Gamma{[\Gamma,\Gamma]}}{\cong}\frac{\Gamma\cap[G,G]}{[\Gamma,\Gamma]}\oplus\mathbb{Z}^2.\tag{21}\] The group of unitary characters of \(\Gamma\), thus, is a finite union of 2-tori, \[\label{E:Gamma42} \hom\bigl(\Gamma,U(1)\bigr)\cong A\times U(1)\times U(1),\tag{22}\] where \(A=\hom\left(\frac{\Gamma\cap[G,G]}{[\Gamma,\Gamma]},U(1)\right)\) is a finite abelian group.

To specify a character \(\chi\colon\Gamma\to U(1)\) it suffices to know its values on the generators, \(\chi(\gamma_i)\) for \(i=1,\dotsc,5\).

3 Decomposition into irreducible representations↩︎

If \(\Gamma\) is a lattice in a simply connected nilpotent Lie group \(N\), then \(L^2(\Gamma\setminus N)\) decomposes into a countable direct sum of irreducible unitary representations of \(N\). The multiplicities of the representations appearing in this decomposition have been studied by Moore [48]. An explicit formula for these multiplicities has been conjectured by Mostow and proved, independently, by Howe [45] and Richardson [46]. More generally, for every unitary character \(\chi\colon\Gamma\to U(1)\), the induced representation \[L^2\bigl(N\times_\chi\mathbb{C}\bigr) =\left\{g\colon N\to\mathbb{C}\, \middle|\begin{array}{c}\text{g(\gamma n)=\chi(\gamma)g(n) for \gamma\in\Gamma and n\in N,}\\|g|\in L^2(\Gamma\setminus N)\end{array} \right\}\] decomposes into a countable direct sum of irreducible unitary representations and the multiplicities are known explicitly, cf. [45] and [46].

Let us briefly recall some of the terminology used in the preceding lemma. To this end let \(\mathfrak n\) denote the Lie algebra of \(N\), suppose \(f\in\mathfrak n^*\), and let \(\mathfrak m\) denote a maximal subordinate subalgebra, i.e., a subalgebra of maximal dimension in \(\mathfrak n\) such that \(f([\mathfrak m,\mathfrak m])=0\). Then \[\label{E:barf} \bar f(m):=e^{2\pi\mathbf{i}f(\log m)}\tag{26}\] defines a unitary character on the group \(M:=\exp(\mathfrak m)\). According to Kirillov [49]–[51] such a character induces an irreducible unitary representation of \(N\) given by right translation on the Hilbert space \[\left\{h\colon N\to\mathbb{C}\, \middle|\begin{array}{c}\text{h(mn)=\bar f(m)h(n) for m\in M and n\in N,}\\|h|\in L^2(M\setminus N)\end{array} \right\}.\] One may always assume \(\mathfrak m\) to be special [46] and then \((\bar f,M)\) is called a maximal character [46]. The group \(N\) acts by conjugation on the set of maximal characters [46] and the stabilizer of \((\bar f,M)\) is \(M\), see [46]. A maximal character \((\bar f,M)\) is called rational [46] if it can be obtained from a (rational) linear functional \(f\in\mathfrak n^*\) mapping \(\log\Gamma\) into the rational numbers and \(\mathfrak m\) is rational with respect to the rational structure on \(\mathfrak n\) provided by \(\log\Gamma\). Note that \(\mathfrak m\) is rational if and only if \(\Gamma\cap M\) is a lattice in \(M\), i.e., if and only if \((\Gamma\cap M)\setminus M\) is compact. Clearly, the action of \(\Gamma\) preserves the subset of rational maximal characters and the integrality condition \(\bar f|_{\Gamma\cap M}=\chi|_{\Gamma\cap M}\).

In the remaining part of this section we will specialize the preceding lemma to the 5-dimensional Lie group \(G\) considered before. All irreducible unitary representations of this Lie group have been described explicitly by Dixmier in [52]. There are three types of irreducible unitary representations of \(G\) which we now describe in terms of a graded basis \(X_1,\dotsc,X_5\) of \(\mathfrak g\) satisfying 1 . If \(\rho\) is a representation of \(G\), we let \(\rho'\) denote the corresponding infinitesimal representation of \(\mathfrak g\).

(I) Scalar representations: For \((\alpha,\beta)\in\mathbb{R}^2\) there is an irreducible unitary representation \(\rho_{\alpha,\beta}\) of \(G\) on \(\mathbb{C}\) such that \[\label{E:rep46scalar} \rho'_{\alpha,\beta}(X_1)=2\pi\mathbf{i}\alpha,\qquad \rho'_{\alpha,\beta}(X_2)=2\pi\mathbf{i}\beta,\tag{27}\] and \(\rho'_{\alpha,\beta}(X_3)=\rho'_{\alpha,\beta}(X_4)=\rho'_{\alpha,\beta}(X_5)=0\). Via Kirillov’s construction, the functional \(f\in\mathfrak g^*\) with \(f(X_1)=\alpha\), \(f(X_2)=\beta\), and \(f(X_3)=f(X_4)=f(X_5)=0\) induces a representation isomorphic to \(\rho_{\alpha,\beta}\). These are precisely the irreducible representations which factor through the abelianization \(G/[G,G]\cong\mathbb{R}^2\).

(II) Schrödinger representations: For \(0\neq\hbar\in\mathbb{R}\) there is an irreducible unitary representation \(\rho_\hbar\) of \(G\) on \(L^2(\mathbb{R})=L^2(\mathbb{R},d\theta)\) such that \[\label{E:Schroedinger} \rho'_\hbar(X_1)=\partial_\theta,\qquad \rho'_\hbar(X_2)=2\pi\mathbf{i}\hbar\cdot\theta,\qquad \rho'_\hbar(X_3)=2\pi\mathbf{i}\hbar,\tag{28}\] and \(\rho'_\hbar(X_4)=\rho'_\hbar(X_5)=0\). Via Kirillov’s construction, any functional \(f\in\mathfrak g^*\) with \(f(X_3)=\hbar\) and \(f(X_4)=f(X_5)=0\) induces a representation isomorphic to \(\rho_\hbar\). These are precisely the irreducible unitary representations which factor through the 3-dimensional Heisenberg group \(H=G/Z\) but do not factor through the abelianization \(G/[G,G]\).

(III) Generic representations: For real numbers \(\lambda,\mu,\nu\) with \((\lambda,\mu)\neq(0,0)\) there is an irreducible unitary representation \(\rho_{\lambda,\mu,\nu}\) of \(G\) on \(L^2(\mathbb{R})=L^2(\mathbb{R},d\theta)\) such that: \[\begin{align} \notag \rho'_{\lambda,\mu,\nu}(X_1) &=\frac{\lambda{(\lambda^2+\mu^2)^{1/3}}}{\cdot}\partial_\theta-\frac{2\pi\mathbf{i}\mu}{(\lambda^2+\mu^2)^{1/3}}\cdot\frac{\theta^2+\nu(\lambda^2+\mu^2)^{-2/3}}{2} \\\notag \rho'_{\lambda,\mu,\nu}(X_2) &=\frac{\mu{(\lambda^2+\mu^2)^{1/3}}}{\cdot}\partial_\theta+\frac{2\pi\mathbf{i}\lambda}{(\lambda^2+\mu^2)^{1/3}}\cdot\frac{\theta^2+\nu(\lambda^2+\mu^2)^{-2/3}}{2} \\\label{E:rep46gen46X3} \rho'_{\lambda,\mu,\nu}(X_3) &=2\pi\mathbf{i}(\lambda^2+\mu^2)^{1/3}\cdot\theta, \\\notag \rho'_{\lambda,\mu,\nu}(X_4) &=2\pi\mathbf{i}\lambda, \\\notag \rho'_{\lambda,\mu,\nu}(X_5) &=2\pi\mathbf{i}\mu. \end{align}\tag{29}\] This differs from the representation given in [52] by a conjugation with a unitary scaling on \(L^2(\mathbb{R})\) which we have introduced for better compatibility with the grading automorphism. Note that \[\rho'_{\lambda,\mu,\nu}\bigl(X_3X_3+2X_1X_5-2X_2X_4\bigr)=(2\pi)^2\nu.\] Any functional \(f\in\mathfrak g^*\) with \(f(X_4)=\lambda\), \(f(X_5)=\mu\), and \[\label{E:rep46gen46nu} f(X_3)^2+2f(X_1)f(X_5)-2f(X_2)f(X_4)=\nu\tag{30}\] induces a representation isomorphic to \(\rho_{\lambda,\mu,\nu}\).

The representations listed above are mutually nonequivalent, and they comprise all equivalence classes of irreducible unitary representations of \(G\). Given the convention in 26 it turns out to be convenient to incorporate the factors \(2\pi\) in the labeling of the representations, as indicated above, when considering integrality with respect to a lattice \(\Gamma\) generated by the exponentials of \(\tilde{\gamma}_i\) as in 10 .

For integers \(l,r,w,n\in\mathbb{Z}\) with \(l,r\geq1\) we define \[\label{E:mult} m(l,r,w,n):=\sharp\Bigl\{k\in\mathbb{Z}/l\mathbb{Z}\Bigm|rk(k+l)+2wk\equiv n\mod2l\mathbb{Z}\Bigr\}.\tag{38}\] Clearly, \[\label{E:M46mult} m(l,r,w,n+2l)=m(l,r,w,n)\tag{39}\] and \[\label{E:m46total} \sum_{n=1}^{2l}m(l,r,w,n)=l.\tag{40}\]

With this notation, the statement in Lemma 5 may be expressed in the form: \[\begin{gather} \label{E:rep46deco} L^2(G\times_\chi\mathbb{C}) =\underbrace{\bigoplus_{(\alpha,\beta)\in(a,b)+\mathbb{Z}^2}\rho_{\alpha,\beta}}_{\substack{\textrm{only appears if}\\\textrm{\chi|_{\Gamma\cap[G,G]}=1}}} \,\,\,\oplus\,\,\,\underbrace{\bigoplus_{\hbar\in c+r\mathbb{Z}}|\hbar|\rho_\hbar}_{\substack{\textrm{only appears if}\\\textrm{\chi|_{\Gamma\cap Z}=1}}} \\ \oplus\bigoplus_{\substack{(0,0)\neq(\lambda,\mu)\in\\(\lambda_0,\mu_0)+(\Gamma'')^*}}\quad\bigoplus_{n\in\mathbb{Z}}\,\,\,m\bigl(\tfrac dr,r,w,n\bigr)\rho_{\lambda,\mu,\nu_0+rn}. \end{gather}\tag{41}\] Here \(\lambda_0,\mu_0\in r\mathbb{Z}\) are as in Lemma 3, and \[\label{E:Gamma393942} (\Gamma'')^*=\left\{(l,m)\in\mathbb{R}^2\middle|\begin{array}{c}\frac{l}{r}\in\mathbb{Z},\frac{m}{r}\in\mathbb{Z},l\frac{u-1}{2}+m\frac{v-1}{2}\in\mathbb{Z}\\le+mf\in\mathbb{Z},lg+mh\in\mathbb{Z}\end{array} \right\} \subseteq(r\mathbb{Z})\times(r\mathbb{Z})\tag{42}\] denotes the lattice dual to the lattice \(\Gamma''\) spanned by the vectors in 11 . Moreover, \(d=\gcd(\lambda,\mu)\), and \(w\), \(\nu_0\) are defined 34 and 35 .

4 Decomposition of the zeta function↩︎

In this section we use the decomposition in 41 to decompose the twisted Rumin complex associated with a standard (2,3,5) distribution on the nilmanifold \(\Gamma\setminus G\), a standard fiberwise graded Euclidean inner product on \(\mathfrak t(\Gamma\setminus G)\), and a unitary character \(\chi\colon\Gamma\to U(1)\). This yields a decomposition of the corresponding zeta function, cf. 8 .

We continue to use a graded basis \(X_1,\dotsc,X_5\) of \(\mathfrak g\) satisfying the relations in 1 . Let \(\mathcal{D}_G\) denote the left invariant distribution on \(G\) spanned by \(X_1\) and \(X_2\). Left translation provides a trivialization of the tangent bundle that induces a trivialization of the bundle of osculating algebras, \(\mathfrak tG=G\times\mathfrak g\). Passing to the fiberwise Lie algebra cohomology, we obtain a trivialization of the vector bundle \(\mathcal{H}^q(\mathfrak tG)=G\times H^q(\mathfrak g)\). Via this identification, the untwisted Rumin differential is a left invariant differential operator \(D_q\colon C^\infty(G)\otimes H^q(\mathfrak g)\to C^\infty(G)\otimes H^{q+1}(\mathfrak g)\) which may be considered as \[\label{E:Dq46Ug} D_q\in\mathcal{U}(\mathfrak g)\otimes L\bigl(H^q(\mathfrak g),H^{q+1}(\mathfrak g)\bigr)\tag{43}\] where \(\mathcal{U}(\mathfrak g)\) denotes the universal enveloping algebra of \(\mathfrak g\). With respect to a particular basis of \(H^q(\mathfrak g)\), the Rumin differentials take the form: \[\begin{align} D_0&=\begin{pmatrix}X_1\\ X_2\end{pmatrix} \\ D_1&=\begin{pmatrix}-X_{112}-X_{13}-X_4&X_{111}\\-\sqrt2X_{122}-\sqrt2X_5&\sqrt2X_{211}-\sqrt2X_4\\-X_{222}&X_{221}-X_{23}-X_5\end{pmatrix} \\ D_2&=\begin{pmatrix}-X_{12}-X_3&X_{11}/\sqrt2&0\\-X_{22}/\sqrt2&-\tfrac32X_3&X_{11}/\sqrt2\\0&-X_{22}/\sqrt2&X_{21}-X_3\end{pmatrix} \\ D_3&=\begin{pmatrix}X_{122}+X_{32}-X_5&-\sqrt2X_{112}+\sqrt2X_4&X_{111}\\X_{222}&-\sqrt2X_{221}-\sqrt2X_5&X_{211}-X_{31}+X_4\end{pmatrix} \\ D_4&=\begin{pmatrix}-X_2,X_1\end{pmatrix} \end{align}\] Here we are using the notation \(X_{j_1\cdots j_k}=X_{j_1}\cdots X_{j_k}\). These formulas for \(D_q\) have been derived in [53], where they are expressed with respect to a slightly different basis, see also [39] or [30].

Suppose \(\Gamma\) is a lattice in \(G\) that is of the form considered in Section 2, i.e., generated by the exponentials of \(\tilde{\gamma}_i\) as in 10 . Furthermore, let \(\chi\colon\Gamma\to U(1)\) be a unitary character. For the sections of the associated flat line bundle \(F_\chi\) over the nilmanifold \(\Gamma\setminus G\) we have a canonical identification \[\label{E:F46chi46sec} \Gamma^\infty(F_\chi) =C^\infty(G\times_\chi\mathbb{C}) :=\left\{f\in C^\infty(G,\mathbb{C})\middle|\begin{array}{c}f(\gamma g)=\chi(\gamma)f(g)\\\text{for \gamma\in\Gamma and g\in G}\end{array}\right\}.\tag{44}\] The left invariant 2-plane field \(\mathcal{D}_G\) descends to a generic distribution of rank two on the nilmanifold \(\Gamma\setminus G\) which we denote by \(\mathcal{D}_{\Gamma\setminus G}\). The trivialization \(\mathcal{H}^q(\mathfrak tG)=G\times H^q(\mathfrak g)\) mentioned above gives rise to a trivialization of the vector bundle \(\mathcal{H}^q(\mathfrak t(\Gamma\setminus G))=(\Gamma\setminus G)\times H^q(\mathfrak g)\). Combining this with 44 we obtain a canonical identification \[\label{E:HqF46G} \Gamma^\infty\bigl(\mathcal{H}^q(\mathfrak t(\Gamma\setminus G))\otimes F_\chi\bigr)=C^\infty(G\times_\chi\mathbb{C})\otimes H^q(\mathfrak g).\tag{45}\] Via this identification the twisted Rumin differentials become operators \[\label{E:DqG} D_q\colon C^\infty(G\times_\chi\mathbb{C})\otimes H^q(\mathfrak g)\to C^\infty(G\times_\chi\mathbb{C})\otimes H^{q+1}(\mathfrak g)\tag{46}\] which are given by the matrices in 43 , with the universal algebra now acting in the (induced) representation \(C^\infty(G\times_\chi\mathbb{C})\).

Let \(g\) denote the graded Euclidean inner product on \(\mathfrak g\) that turns \(X_1,\dotsc,X_5\) into an orthonormal basis. The corresponding left invariant fiberwise graded Euclidean inner product on the bundle of osculating algebras \(\mathfrak tG=G\times\mathfrak g\) descends to a fiberwise graded Euclidean inner product \(g_{\Gamma\setminus G}\) on the bundle of osculating algebras \(\mathfrak t(\Gamma\setminus G)\) over \(\Gamma\setminus G\). The flat line bundle \(F_\chi\) comes with a canonical fiberwise Hermitian inner product denoted by \(h_\chi\). The metrics \(g_{\Gamma\setminus G}\) and \(h_\chi\) provide an \(L^2\) inner product on the space \(\Gamma^\infty\bigl(\mathcal{H}^q(\mathfrak t(\Gamma\setminus G))\otimes F_\chi\bigr)\). Via the identification in 45 this inner product corresponds to the tensor product of the standard \(L^2\) inner product on the induced representation \(C^\infty(G\times_\chi\mathbb{C})\) and the Hermitian inner product on \(H^q(\mathfrak g)\) induced by \(g\).

In view of 45 , the decomposition of the induced representation described in 41 provides a decomposition of the twisted Rumin complex over \(\Gamma\setminus G\) into a countable orthogonal direct sum of Rumin complexes, each associated with an irreducible unitary representations of \(G\). For every irreducible unitary representation \(\rho\) of \(G\), we let \(\rho(D_q)\) denote the operator induced by 43 in this representation, and consider the corresponding Rumin–Seshadri operator, \[\label{E:Delta46rho} \Delta_{\rho,q}:=\bigl(\rho(D_{q-1})\rho(D_{q-1})^*\bigr)^{a_{q-1}}+\bigl(\rho(D_q)^*\rho(D_q)\bigr)^{a_q}.\tag{47}\] Moreover, we put \[\label{E:zeta46rho46def} \zeta_\rho(s) :=\mathop{\mathrm{str}}\bigl(N\Delta_\rho^{-s}\bigr) :=\sum_{q=0}^5(-1)^qN_q\mathop{\mathrm{tr}}\Delta_{\rho,q}^{-s}.\tag{48}\] These zeta functions are known to converge for \(\Re s\) sufficiently large, depending on \(\rho\), and they admit analytic continuation to meromorphic functions on the entire complex plane which are holomorphic at \(s=0\), see [30]. As the Rumin complex is a Rockland complex, the operators \(\Delta_{\rho,q}\) have trivial kernel for nontrivial \(\rho\). Clearly, \(\Delta_{\rho,q}\) vanishes if \(\rho\) is the trivial representation, and so does \(\zeta_\rho(s)\).

Using the notation from Lemma 5 we define \[\label{E:zeta46I} \zeta_{\mathrm{I},\Gamma,\chi}(s) :=\sideset{}{'}\sum_{(\alpha,\beta)\in(a,b)+\mathbb{Z}^2}\zeta_{\rho_{\alpha,\beta}}(s)\tag{49}\] if \(\chi|_{\Gamma\cap[G,G]}=1\), and \(\zeta_{\mathrm{I},\Gamma,\chi}(s):=0\) otherwise. Moreover, \[\label{E:zeta46II} \zeta_{\mathrm{I\!I},\Gamma,\chi}(s) :=\sideset{}{'}\sum_{\hbar\in c+r\mathbb{Z}}|\hbar|\zeta_{\rho_\hbar}(s)\tag{50}\] if \(\chi|_{\Gamma\cap Z}=1\), and \(\zeta_{\mathrm{I\!I},\Gamma,\chi}(s):=0\) otherwise. Finally, \[\label{E:zeta46III} \zeta_{\mathrm{I\!I\!I},\Gamma,\chi}(s) :=\sideset{}{'}\sum_{\substack{(\lambda,\mu)\in\\(\lambda_0,\mu_0)+(\Gamma'')^*}}\,\,\,\sum_{n\in\mathbb{Z}}\,\,\,m(\tfrac dr,r,w,n)\zeta_{\rho_{\lambda,\mu,\nu_0+nr}}(s).\tag{51}\] Here \(\lambda_0,\mu_0\) are as in Lemma 3, \((\Gamma'')^*\) is the dual lattice in 42 , \(d=\gcd(\lambda,\mu)\), \(w\) is given in 35 , \(\nu_0\) is given in 34 , and the multiplicity is defined in 38 . Moreover, we are using the common convention to decorate the summation symbol with a prime to indicate that the summand with index zero (if any) is omitted.

Subsequently, we will analyze each of the three summands in 52 individually.

5 Evaluation of the zeta function↩︎

We continue to use the notation set up in the preceding section and Lemma 5. In particular, \(\Gamma\) denotes the lattice in \(G\) spanned by the exponentials of \(\tilde{\gamma}_i\) as in 10 , \(\chi\colon\Gamma\to U(1)\) is a unitary character, and \(g\) is a graded Euclidean inner product on \(\mathfrak g\) such that \(X_1,\dotsc,X_5\) is an orthonormal basis of \(\mathfrak g\). This implies, that every graded automorphism \(\phi\in\mathop{\mathrm{Aut}}_\textrm{gr}(\mathfrak g)=\mathop{\mathrm{GL}}(\mathfrak g_{-1})\cong\mathop{\mathrm{GL}}_2(\mathbb{R})\) that preserves \(g|_{\mathfrak g_{-1}}\) also preserves \(g\). By naturality of the Rumin differentials, \(\phi\cdot D_q=D_q\) where the dot denotes the natural left action of \(\mathop{\mathrm{Aut}}_\textrm{gr}(\mathfrak g)\) on \(\mathcal{U}(\mathfrak g)\otimes L\bigl(H^q(\mathfrak g),H^{q+1}(\mathfrak g)\bigr)\), cf. 43 . Hence, if \(\rho\) is an irreducible unitary representation of \(G\), then \[\label{E:zeta46phi} \zeta_{\rho\circ\phi}(s)=\zeta_\rho(s)\tag{55}\] for every graded automorphism \(\phi\in\mathop{\mathrm{Aut}}_\textrm{gr}(\mathfrak g)\) preserving \(g|_{\mathfrak g_{-1}}\). Moreover, for \(\tau>0\) we have \[\label{E:zeta46phit} \zeta_{\rho\circ\phi_\tau}(s)=\tau^{2\kappa s}\zeta_{\rho}(s)\tag{56}\] where \(\phi_\tau\in\mathop{\mathrm{Aut}}_\textrm{gr}(\mathfrak g)\) denotes the grading automorphism acting by \(\tau^j\) on \(\mathfrak g_j\). For more details we refer to [30].

6 Proof of the main theorem↩︎

In order to prove the theorem formulated in the introduction, note first that we may w.l.o.g.assume \(\Gamma\), \(\mathcal{D}_{\Gamma\setminus G}\), and \(g_{\Gamma\setminus G}\) to be of standard form. Indeed, by Lemma 2, we may assume that the lattice \(\Gamma\) is of the form considered in Section 2, i.e., generated by the exponentials of \(\tilde{\gamma}_i\) as in 10 . Moreover, according to [17] the torsion of the Rumin complex does not depend on the choice of \(\mathcal{D}_{\Gamma\setminus G}\) and \(g_{\Gamma\setminus G}\), as long as they are induced from a left invariant (2,3,5) distribution on \(G\) and a left invariant fiberwise graded Euclidean inner product on \(\mathfrak tG\), respectively. Hence we may also assume that \(\mathcal{D}_{\Gamma\setminus G}\) and \(g_{\Gamma\setminus G}\) are of the form considered in Section 4, i.e., \(\mathcal{D}_{\Gamma\setminus G}\) is induced by the left invariant 2-plane field \(\mathcal{D}_G\) on \(G\) spanned by \(X_1\) and \(X_2\), and \(g_{\Gamma\setminus G}\) is induced from a graded Euclidean inner product \(g\) on \(\mathfrak g\) such that \(X_1,\dotsc,X_5\) are orthonormal. Strictly speaking, [17] only covers graded Euclidean inner products induced from left invariant sub-Riemannian metrics on \(\mathcal{D}_G\), but the proof there can readily be extended to the generality required here.

As the Rumin complex computes the de Rham cohomology \(H^*(\Gamma\setminus G;F_\chi)\), one can use the Leray–Serre spectral sequence to prove the acyclicity of the Rumin complex, cf. Lemma 11 below. However, the acyclicity can also be read off the decomposition provided in Lemma 5. Indeed, the irreducible unitary representations appearing in this decomposition are all nontrivial, as \(\chi\) is assumed to be nontrivial. Since the Rumin complex is a Rockland complex, it becomes exact in every nontrivial unitary representation of \(G\). The acyclicity of the Rumin complex thus follows from the decomposition in 53 .

Combining Lemma 6 with Equations 58 , 63 , and 79 yields \[\zeta_{\Gamma\setminus G,\mathcal{D}_{\Gamma\setminus G},F_\chi,g_{\Gamma\setminus G},h_\chi}'(0) =\zeta_{\mathrm{I},\Gamma,\chi}'(0)+\zeta_{\mathrm{I\!I},\Gamma,\chi}'(0)+\zeta_{\mathrm{I\!I\!I},\Gamma,\chi}'(0)=0.\] Using 7 we obtain \[\tau(\Gamma\setminus G,\mathcal{D}_{\Gamma\setminus G},F_\chi,g_{\Gamma\setminus G},h_\chi)=1.\] This completes the proof of the theorem.

To derive the corollary we need to know the Ray–Singer torsion of \(\Gamma\setminus G\).

For a unitary representation \(\rho\colon\Gamma\to U(k)\) we consider the positive real number \[R(\Gamma,\rho):=\frac{\|-\|_{\mathcal{D}_{\Gamma\setminus G},g_{\Gamma\setminus G},h_\rho}^{\mathop{\mathrm{sdet}}H^*(\Gamma\setminus G;F_\rho)}}{\|-\|_\textrm{RS}^{\mathop{\mathrm{sdet}}H^*(\Gamma\setminus G;F_\rho)}}.\] If \(\rho\) is acyclic, that is, if \(H^*(\Gamma\setminus G;F_\rho)=0\), then \(R(\Gamma,\rho)=\frac{\tau_\textrm{RS}(\Gamma\setminus G;F_\rho)}{\tau(\Gamma\setminus G,\mathcal{D}_{\Gamma\setminus G},F_\rho,g_{\Gamma\setminus G},h_\rho)}\), by the very definition. Hence, combining the theorem with Lemma 11 we obtain \[\label{E:R46G46chi} R(\Gamma,\chi)=1\tag{103}\] for every nontrivial character \(\chi\colon\Gamma\to U(1)\). According to [17] the quantity \(R(\Gamma,\chi)\) depends continuously on \(\chi\). Hence, 103 remains true for all unitary characters \(\chi\).

If \(\rho\colon\Gamma\to U(k)\) is irreducible then there exists a sublattice \(\Gamma'\) in \(\Gamma\) and a unitary character \(\chi'\colon\Gamma'\to U(1)\) such that \(\rho\) is isomorphic to the representation of \(\Gamma\) induced by \(\chi'\), see [57]. Hence, \(R(\Gamma,\rho)=R(\Gamma',\chi')\) by [17] and 103 yields \[\label{E:R46G46rho} R(\Gamma,\rho)=1\tag{104}\] for irreducible \(\rho\). Clearly, \(R(\Gamma,\rho_1\oplus\rho_2)=R(\Gamma,\rho_1)\cdot R(\Gamma,\rho_2)\) for any two unitary representations \(\rho_1\) and \(\rho_2\) of \(\Gamma\), see [17]. Hence, 104 remains true for all finite dimensional unitary representation \(\rho\) of \(\Gamma\). This completes the proof of the corollary.

Acknowledgments↩︎

This research was funded in whole or in part by the Austrian Science Fund (FWF) Grant DOI 10.55776/P31663.

References↩︎