The Fried conjecture in small dimensions

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Abstract

We study the twisted Ruelle zeta function \(\zeta _X(s)\) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove the Fried conjecture, relating Reidemeister torsion and \(\zeta _X(0)\). In higher dimensions, we show more generally that \(\zeta _X(0)\) is locally constant with respect to the vector field X under a spectral condition. As a consequence, we also show the Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where the Fried conjecture holds true.

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Notes

  1. 1.

    Recall from [27] that \(\tau _{\rho }(M)^2=\tau _{\tilde{\rho }}(\mathcal {M})\).

  2. 2.

    Observe that \(E'\) can be identified with E via the Hermitian structure.

  3. 3.

    Even if not explicitely written in [21], this observation can be deduced from paragraph 3.2 of this reference and from Lemma 3.2 above.

  4. 4.

    Similar method is also used in [22] for Selberg zeta function on surfaces of constant curvature.

  5. 5.

    As we shall consider families \(\tau \mapsto X_{\tau }\), if no confusion is possible we will use the index (or the exponent) \(\tau \) instead of \(X_{\tau }\) in the various quantities \(\varphi _t^{X_{\tau }}\), \(\zeta _{X_{\tau },\rho }\), etc.

  6. 6.

    For a nonprimitive orbit \(k.\gamma \), this is equal to \(\varepsilon _{k.\gamma }=\varepsilon _{\gamma }^k\).

  7. 7.

    The operator \({\hat{\chi }}_1^*{\hat{\chi }}_1\) is not necessary for this proof but will be useful for the wavefront set analysis later.

  8. 8.

    This implies that the Lagrangian states \((f_h)_{0<h\le 1}\) verifies \(\mathrm{WF}_h(f_h)\subset U\) [69, p. 190].

  9. 9.

    We can use [18, Lemma E.45] to justify the convergence in the inequality.

  10. 10.

    This follows from surjectivity of the map \(u\in C^\infty (\mathcal {M};\mathcal {E}_0^{n-k-2})\mapsto u\wedge d\alpha \in C^\infty (\mathcal {M};\mathcal {E}_0^{n-k})\).

  11. 11.

    The proof is even simpler in this case as adding \(is\alpha _0(X_0)\) only modifies the operator by a subsprincipal symbol.

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Acknowledgements

We would like to thank V. Baladi for pointing out to us the papers [62, 63] which are used in the last part of Theorem 1. We also would like to thank Y. Bonthonneau, N.T. Dang, P. Dehornoy, F. Faure, S. Gouëzel, B. Hasselblatt, B. Kuester, F. Naud, H. H. Rugh, H. Sanchez-Morgado, T. Weich for discussions, answers to our questions and crucial remarks on this project. We also thank the referees for their detailed and useful comments that helped us to improve the presentation of our proofs. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 725967). CG and GR were partially supported by the ANR project GERASIC (ANR-13-BS01-0007-01) and GR also acknowledges the support of the Labex CEMPI (ANR-11-LABX-0007-01).

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Appendices

Appendix A: Proof of Lemma 3.2

A.1. Family of order functions

In this paragraph, we fix the aperture of the cones \(\alpha _0>0\) small enough to ensure that \(C^{ss}(\alpha _0)\cap C^u(\alpha _0)=\emptyset \) and we fix some small parameter \(\delta >0\). We construct an order function for every X in a small enough neighborhood of \(X_0\). For that purpose, we closely follow the lines of [21, Lemma 2.1]. We fix \(T_{\alpha _0}'>1\)\(T_{\alpha _0}\) is given by Lemma 3.1. The time \(T_{\alpha _0}'\) will be determined later on in a way that depends only on \(\alpha _0\). For our construction, we also let \(m_0(x,\xi )\in \mathcal {C}^{\infty }(S^*\mathcal {M},[0,1])\) to be equal to 1 on \(C^{u}(\alpha _0)\) and to 0 on \(C^{ss}(\alpha _0)\). Then, we set

$$\begin{aligned} m_X(x,\xi ):=\frac{1}{2T_{\alpha _0}'}\int _{-T_{\alpha _0}'}^{T_{\alpha _0}'}m_0\circ \tilde{\Phi }^{X}_t(x,\xi )dt. \end{aligned}$$
(A.1)

Note that \(m_X\)depends smoothly onX as we chose \(T_{\alpha _0}'\) independently of X near \(X_0\). First of all, we note that

$$\begin{aligned} {\tilde{X}}_Hm_X(x,\xi )=\frac{1}{2T_{\alpha _0}'}\left( m_0\circ \tilde{\Phi }^X_{T_{\alpha _0}'}(x,\xi )- m_0\circ \tilde{\Phi }^X_{-T_{\alpha _0}'}(x,\xi )\right) , \end{aligned}$$
(A.2)

where \({\tilde{X}}_H\) is the vector field of \(\tilde{\Phi }^X_{t}.\) We also observe that, for every \((x,\xi )\) inside \(S^*\mathcal {M}\), the set

$$\begin{aligned} \mathcal {I}_{X_0}(x,\xi ):=\left\{ t\in {\mathbb {R}}:\tilde{\Phi }^{X_0}_t(x,\xi )\in S^*\mathcal {M}\backslash \left( C^u(\alpha _0/2)\cup C^{ss}(\alpha _0/2)\right) \right\} \end{aligned}$$

is an interval whose length is bounded by some constant \(T_{\alpha _0}''>0\). Fix now a point \((x,\xi )\in S^*{\tilde{M}}\) and a vector field which is close enough to \(X_0\) (to be determined). If \(\tilde{\Phi }^X_t(x,\xi )\in C^u(\alpha _0)\) for every \(t\in {\mathbb {R}}\), then the set

$$\begin{aligned} \tilde{\mathcal {I}}_{X}(x,\xi ):=\left\{ t\in {\mathbb {R}}:\tilde{\Phi }^{X}_t(x,\xi )\in S^*\mathcal {M}\backslash \left( C^u(\alpha _0)\cup C^{ss}(\alpha _0)\right) \right\} \end{aligned}$$

is empty and the same holds if \(\tilde{\Phi }^X_t(x,\xi )\in C^{ss}(\alpha _0)\) for every \(t\in {\mathbb {R}}\). Hence, it remains to bound the length of \(\tilde{\mathcal {I}}_{X}(x,\xi )\) when the orbit of \((x,\xi )\) crosses \(S^*\mathcal {M}\backslash \left( C^u(\alpha _0)\cup C^{ss}(\alpha _0)\right) \) and we may suppose without loss of generality that \((x,\xi )\in S^*\mathcal {M}\backslash \left( C^u(\alpha _0)\cup C^{ss}(\alpha _0)\right) \). Up to the fact that we may have to decrease a little bit the size of the set \(\mathcal {U}_{\alpha _0}(X_0)\) appearing in Lemma 3.1, we have that \(\tilde{\Phi }_{T_{\alpha _0}''}^{X}(x,\xi )\) belongs to \(C^{u}(\alpha _0)\). Hence, thanks to Lemma 3.1, one finds that, for every \(t\ge T_{\alpha _0}''+1\), one has \(\tilde{\Phi }_t^X(x,\xi )\in C^u(\alpha _0)\). The same holds in backward times. Hence, the diameter of \(\tilde{\mathcal {I}}_{X}(x,\xi )\) is uniformly bounded by \(2(1+T_{\alpha _0}'')\) and we pick \(T_{\alpha _0}'=\frac{1+T_{\alpha _0}''}{\delta }\) for \(\delta <1\).

We set

$$\begin{aligned} \mathcal {O}^{u}(X)=\tilde{\Phi }^{X}_{T_{\alpha _0}'}(S^*\mathcal {M}\backslash C^{ss}(\alpha _0))\ \text {and}\ \mathcal {O}^{ss}(X)=\tilde{\Phi }^{X}_{-T_{\alpha _0}'} (S^*\mathcal {M}\backslash C^{u}(\alpha _0)). \end{aligned}$$

Let us now discuss the properties of \(m_X\) for X belonging to \(\mathcal {U}_{\alpha _0}(X_0)\):

  1. (1)

    If \((x,\xi )\in \mathcal {O}^u(X)\), then \(\tilde{\Phi }^X_{-T_{\alpha _0}'}(x,\xi )\notin C^{ss}(\alpha _0)\). Hence, from the definition of \(T_{\alpha _0}'\), one has \(\tilde{\Phi }^X_{T_{\alpha _0}'}(x,\xi )\in C^{u}(\alpha _0)\) and, from (A.2), one deduce that \({\tilde{X}}_Hm_X\ge 0\) on \(\mathcal {O}^u(X)\). Similarly, one has

    $$\begin{aligned} m_X(x,\xi )= & {} \frac{1}{2T_{\alpha _0}'}\left( \int _{-T_{\alpha _0}'}^{-T_{\alpha _0}'+2(T_{\alpha _0}+T_{\alpha _0}'')}m_0\circ \tilde{\Phi }^{X}_t(x,\xi )dt\right. \\&\left. + \int _{-T_{\alpha _0}'+2(T_{\alpha _0}+T_{\alpha _0}'')}^{T_{\alpha _0}'}m_0\circ \tilde{\Phi }^{X}_t(x,\xi )dt\right) , \end{aligned}$$

    from which one can infer

    $$\begin{aligned} \forall (x,\xi )\in \mathcal {O}^u(X),\quad m_{X}(x,\xi )\ge 1-\frac{T_{\alpha _0}+T_{\alpha _0}''}{T_{\alpha _0}'}=1-\delta . \end{aligned}$$
  2. (2)

    Reasoning along similar lines, one also finds that, for every \((x,\xi )\in \mathcal {O}^{ss}(X)\), \({\tilde{X}}_Hm_X\ge 0\) and

    $$\begin{aligned} m_{X}(x,\xi )\le \delta . \end{aligned}$$
  3. (3)

    Let \((x,\xi )\) be an element of \(S^*\mathcal {M}\backslash (\mathcal {O}^u(X)\cup \mathcal {O}^{ss}(X)).\) In that case, one has \(\tilde{\Phi }^X_{-T_{\alpha _0}'}(x,\xi )\in C^{ss}(\alpha _0)\) and \(\tilde{\Phi }^X_{T_{\alpha _0}'}(x,\xi )\in C^u(\alpha _0)\). Thus, one finds

    $$\begin{aligned} {\tilde{X}}_Hm_X(x,\xi )= & {} \frac{1}{2T_{\alpha _0}'}\left( m_0\circ \tilde{\Phi }^X_{T_{\alpha _0}'}(x,\xi )-m_0\circ \tilde{\Phi }^X_{-T_{\alpha _0}'}(x,\xi )\right) \nonumber \\= & {} \frac{1}{2T_{\alpha _0}'}>0. \end{aligned}$$
    (A.3)
  4. (4)

    Let now \((x,\xi )\in S^*\mathcal {M}\backslash C^{u}(\alpha _0)\). Write

    $$\begin{aligned} m_X(x,\xi )\le \frac{1}{2}+\frac{1}{2T_{\alpha _0}'} \int _{-T_{\alpha _0}'}^0m_0\circ \tilde{\Phi }^{X}_t(x,\xi )dt\le \frac{1+\delta }{2}. \end{aligned}$$

Let us conclude this construction with the following useful observation:

Lemma A.1

Let \(\alpha _0>0\) be small enough to ensure that \(C^{u}(\alpha _0)\cap C^{ss}(\alpha _0)=\emptyset \). Then, there exists \(0<\alpha _1<\alpha _0\) and a neighborhood \(\mathcal {U}_{\alpha _0}(X_0)\) of \(X_0\) in \(\mathcal {A}\) such that, for every \(X\in \mathcal {U}_{\alpha _0}(X_0)\),

$$\begin{aligned} C^u(\alpha _1)\cap S^*\mathcal {M}\subset \mathcal {O}^u(X)\quad \text {and}\quad C^{ss}(\alpha _1)\cap S^*\mathcal {M}\subset \mathcal {O}^{ss}(X). \end{aligned}$$

Proof

First of all, we note that \(S^*\mathcal {M}\cap C^u(0)\) is invariant by the flow \(\tilde{\Phi }^{X_0}_t\) and it is disjoint from \(C^{ss}(\alpha _0)\cap S^*\mathcal {M}\). Hence, by construction of \(\mathcal {O}^u(X_0)\), one can find some small enough \(\alpha _1>0\) such that \(S^*\mathcal {M}\cap C^u(\alpha _1)\) is contained inside \({\mathcal {O}}^u(X_0)\). By continuity with respect to X, this property remains true for any X close enough to \(X_0\), i.e. \(S^*\mathcal {M}\cap C^u(\alpha _1)\subset {\mathcal {O}}^u(X)\) for any \(X\in \mathcal {U}_{\alpha _0}(X_0)\cap \mathcal {A}.\) The same proof works for the second part of the Lemma. \(\square \)

Remark 11

In all the construction so far, we could have defined the cones \(C^{uu}(\alpha )\) and \(C^s(\alpha )\) (see paragraph 3.1) and a decaying order function \({\tilde{m}}_X(x,\xi )\) which is close to 0 on \(C^{s}(\alpha )\) and close to 1 on \(C^{uu}(\alpha )\).

A.2. Definition of the escape function

We start with the construction of the function \(f(x,\xi )\in \mathcal {C}^{\infty }(T^*M,{\mathbb {R}}_+)\). For \(\Vert \xi \Vert _x\ge 1\), it will be 1-homogeneous and equal to \(\Vert \xi \Vert _x\) outside the cones \(C^{uu}(\tilde{\alpha }_0)\) and \(C^{ss}(\tilde{\alpha }_0)\) for \(\tilde{\alpha }_0>0\) small enough (to be determined). Following the proof of [16, Lemma C.1] (see also [35, Lemma 2.2]), we set, for \((x,\xi )\) near \(C^{ss}(\tilde{\alpha }_0/2)\) and \(\Vert \xi \Vert _x\ge 1\),

$$\begin{aligned} f(x,\xi ):=\exp \left( \frac{1}{T_1}\int _{0}^{T_1}\ln \Vert (d\varphi ^{X_0}_{t}(x)^T)^{-1}\xi \Vert _{\varphi _{X_0}^t(x)}dt\right) . \end{aligned}$$

Recall that, for every \(\xi \) in \(E_s^*(X_0,x)\), one has \(\Vert (d\varphi ^{X_0}_{t}(x)^T)^{-1}\xi \Vert \le Ce^{-\beta t}\Vert \xi \Vert \) for every \(t\ge 0\) (where \(C,\beta \) are some uniform constants). Hence, if we set \(T_1=2\frac{\ln C}{\beta }\), we find that, for every \((x,\xi )\in E_s^*(X_0)\) with \(\Vert \xi \Vert _x\ge 1\), \(X_{H_0}f(x,\xi )\le - f(x,\xi )\frac{\beta }{2}.\) Similarly, picking \(T_1\) large enough, we set, for \((x,\xi )\) near \(C^{uu}(\tilde{\alpha }_0/2)\) and \(\Vert \xi \Vert _x\ge 1\),

$$\begin{aligned} f(x,\xi ):=\exp \left( \frac{1}{T_1}\int _{0}^{T_1}\ln \Vert (d\varphi ^{X_0}_{t}(x)^T)^{-1}\xi \Vert _{\varphi _{X_0}^t(x)}dt\right) , \end{aligned}$$

and we find that \(X_{H_0}f(x,\xi )\ge f(x,\xi )\frac{\beta }{2}\) on \(E_u^*(X_0)\). By continuity, we find that there exists some (small enough) \(\tilde{\alpha }_0>0\) such that, for every \(\Vert \xi \Vert _x\ge 1\),

$$\begin{aligned} (x,\xi )\in C^{ss}(\tilde{\alpha }_0/2)\Rightarrow X_{H_0}f(x,\xi )\le - f(x,\xi )\frac{\beta }{3}, \end{aligned}$$
(A.4)

and

$$\begin{aligned} (x,\xi )\in C^{uu}(\tilde{\alpha }_0/2)\Rightarrow X_{H_0}f(x,\xi )\ge f(x,\xi )\frac{\beta }{3}. \end{aligned}$$
(A.5)

As the function \(f(x,\xi )\) is 1-homogeneous, we can find a neighborhood \(\mathcal {U}(X_0)\) of \(X_0\) in the \(\mathcal {C}^{\infty }\)-topology such that, for every X in \(\mathcal {U}(X_0)\) and for every \(\Vert \xi \Vert _x\ge 1\),

$$\begin{aligned} (x,\xi )\in C^{ss}(\tilde{\alpha }_0/2)\Rightarrow X_{H}f(x,\xi )\le - f(x,\xi )\frac{\beta }{4}, \end{aligned}$$
(A.6)

and

$$\begin{aligned} (x,\xi )\in C^{uu}(\tilde{\alpha }_0/2)\Rightarrow X_{H}f(x,\xi )\ge f(x,\xi )\frac{\beta }{4}. \end{aligned}$$
(A.7)

Finally, we note that there exists some uniform constant \(C>0\) such that, for every X in \(\mathcal {U}(X_0)\) and for \(\Vert \xi \Vert _x\ge 1\),

$$\begin{aligned} -Cf(x,\xi )\le X_Hf(x,\xi )\le C f(x,\xi ) \end{aligned}$$
(A.8)

We are now ready to construct our family of escape functions \(G_X^{N_0,N_1}(x,\xi )\):

$$\begin{aligned} G_X^{N_0,N_1}(x,\xi ):=m_{X}^{N_0,N_1}(x,\xi )\ln (1+f(x,\xi )), \end{aligned}$$

with \(m_{X}^{N_0,N_1}\in \mathcal {C}^{\infty }(T^*M,[-2N_0,2N_1])\) which is 0-homogeneous for \(\Vert \xi \Vert _x\ge 1\). In order to construct this function, we will make use of the order functions defined in paragraph A.1 as in [21, p. 337-8]. Before doing that, let us observe that

$$\begin{aligned} X_HG_X^{N_0,N_1}(x,\xi )= & {} X_H(m_{X}^{N_0,N_1})(x,\xi )\ln (1+f(x,\xi ))\nonumber \\&+\,m_{X}^{N_0,N_1}(x,\xi )\frac{X_Hf(x,\xi )}{1+f(x,\xi )}. \end{aligned}$$
(A.9)

We now fix a small enough neighborhood \(\mathcal {U}(X_0)\) of \(X_0\) so that f enjoys (A.6) and (A.7) for all X in \(\mathcal {U}(X_0)\) and so that we can apply the results of paragraph A.1. Following [21], we set, for \(\Vert \xi \Vert _x\ge 1\),

$$\begin{aligned} m_X^{N_0,N_1}(x,\xi ):= & {} N_1\left( 2-m_X\left( x,\frac{\xi }{\Vert \xi \Vert _x}\right) -{\tilde{m}}_X\left( x,\frac{\xi }{\Vert \xi \Vert _x}\right) \right) \nonumber \\&-\,2N_0{\tilde{m}}_X\left( x,\frac{\xi }{\Vert \xi \Vert _x}\right) , \end{aligned}$$
(A.10)

where we used the conventions of paragraph A.1 and Remark 11. First, notice that, by construction, \(X_H(m_{X}^{N_0,N_1})\le 0\) for \(\Vert \xi \Vert _x\ge 1\). Recall that the order functions \(m_X\) and \({\tilde{m}}_X\) depends on the parameters \(\alpha _0>0\) and \(\delta >0\) and that they depend smoothly on X. Now, we fix \(0<\delta <\frac{1}{2}\min \{1,\min \{N_0,N_1\}/(N_0+N_1)\}\), \(0<16N_0<N_1\) and \(0<\alpha _0<\tilde{\alpha }_0/2\). We then find that \(m_X^{N_0,N_1}(x,\xi /\Vert \xi \Vert _x)\ge N_1\) on \(\mathcal {O}^{ss}(X)\) and \(m_X^{N_0,N_1}(x,\xi /\Vert \xi \Vert _x)\le -N_0\) on \(\mathcal {O}^{uu}(X)\). We also have that \(m_X^{N_0,N_1}(x,\xi /\Vert \xi \Vert _x)\ge \frac{N_1}{4}-2N_0\ge N_1/8\) for \((x,\xi )\) outside \(C^{uu}(\alpha _0)\) (as \(N_1>16N_0\)). We now fix \(\alpha _1\) to be the aperture of the cone appearing in Lemma A.1. This allows to verify the first three requirements of \(m_X^{N_0,N_1}\).

Remark 12

We could also have defined

$$\begin{aligned} {\tilde{m}}_X^{N_0,N_1}(x,\xi ):= N_1\left( 1-m_X\left( x,\frac{\xi }{\Vert \xi \Vert _x}\right) \right) -N_0{\tilde{m}}_X\left( x,\frac{\xi }{\Vert \xi \Vert _x}\right) . \end{aligned}$$

We still have \({\tilde{m}}_X^{N_0,N_1}(x,\xi )\ge N_1\) on \(\mathcal {O}^{ss}(X)\), \({\tilde{m}}_X^{N_0,N_1}(x,\xi )\le \frac{N_1}{4}-N_0\) outside \(C^{ss}(\alpha _0)\).

Finally, combining \(X_H(m_{X}^{N_0,N_1})\le 0\) with (A.9) for \(||\xi ||\ge 1\), we immediately get the upper bound (3.5). It now remains to verify the decay property (3.3). For that purpose, we shall use the conventions of paragraph A.1 and set, for every \(X\in \mathcal {U}(X_0)\),

$$\begin{aligned}&\tilde{\mathcal {O}}^{uu}(X)=\mathcal {O}^{uu}(X)\cap \mathcal {O}^u(X),\ \tilde{\mathcal {O}}^{0}(X)=\mathcal {O}^{s}(X)\cap \mathcal {O}^u(X),\ \text {and}\ \tilde{\mathcal {O}}^{ss}(X)\\&\quad = \mathcal {O}^{ss}(X)\cap \mathcal {O}^s(X), \end{aligned}$$

which contains respectively \(C^{uu}(\alpha _1)\), \(C^u(\alpha _1)\cap C^s(\alpha _1)\) and \(C^{ss}(\alpha _1)\) for \(\alpha _1>0\) small enough (see Lemma A.1). Note also that \(\tilde{\mathcal {O}}^{0}(X)\) is contained inside \(C^u(\alpha _0)\cap C^s(\alpha _0)\) which is a small vicinity of \(E_0^*(X_0)\). Based on (A.9), we can now establish (3.3) except in this small cone around the flow direction. Outside \(\tilde{\mathcal {O}}^{uu}(X)\cup \tilde{\mathcal {O}}^{0}(X)\cup \tilde{\mathcal {O}}^{ss}(X)\), it follows from (A.3) and (A.9). Inside \(\tilde{\mathcal {O}}^{uu}(X)\) and \(\tilde{\mathcal {O}}^{ss}(X)\), it follows from (A.6), (A.7) and (A.9).

Appendix B: Selberg zeta function on trace-free symmetric tensors

Proposition B.1

Let n be even and \(M=\Gamma \backslash {\mathbb {H}}^{n+1}\) be a compact hyperbolic manifold. Let \(\rho :\pi _1(M)\rightarrow U(V_\rho )\) be a finite dimensional unitary representation and let \(\sigma _m\) be the irreducible unitary representation of \(\mathrm{SO}(n)\) into the space \(S_0^m{\mathbb {R}}^n\) of trace-free symmetric tensors of order \(m\ge 1\) on \({\mathbb {R}}^n\). Then the Selberg zeta function \(Z_{S,\sigma _m}(s)\) on M associated to \(\sigma _p\) and \(\rho \) is holomorphic and the order of its zeros are given by

$$\begin{aligned}&\mathrm{ord}_{s_0}Z_{S,\sigma _m}(s)\\&\quad =\left\{ \begin{array}{ll} \dim \ker (\nabla ^*\nabla -n^2/4-m+(s_0-n/2)^2)\cap \ker D^* &{} \text { if }s_0\not =n/2\\ 2\dim \ker (\nabla ^*\nabla -n^2/4-m)\cap \ker D^* &{} \text { if }s_0=n/2 \end{array}\right. \end{aligned}$$

where \(\nabla \) is the twisted Levi-Civita covariant derivative on \(S_0^mT^*M\otimes E\), \(E\rightarrow M\) being the flat bundle over M obtained from the representation \(\rho \), and \(D^*=-\mathrm{Tr}\circ \nabla \) is the divergence operator.

Proof

We follow [6, Theorem 3.15]. First we need to view \(\sigma _m\) as the restriction of a sum of irreducibles representations of \(\mathrm{SO}(n+1)\) as in Section 1.1.2 [6]: it is not difficult to check that

$$\begin{aligned} \sigma _m=(\Sigma _m-\Sigma _{m-1})|_{\mathrm{SO}(n)} \end{aligned}$$

where \(\Sigma _m\) denotes the irreducible unitary representation of \(\mathrm{SO}(n+1)\) into the space \(S_0^m{\mathbb {R}}^{n+1}\). By Section 1.1.3 of [6], there is a \({\mathbb {Z}}^2\)-graded homogeneous vector bundle \(V_{\sigma _m}=V_{\Sigma _m}^+\oplus V_{\Sigma _m}^-\) over \({\mathbb {H}}^{n+1}\) with \(V_{\Sigma _m}^+=S_0^{m}{\mathbb {R}}^{n+1}\) and \(V_{\Sigma _m}^-=S_0^{m-1}{\mathbb {R}}^{n+1}\), and we define the bundle \(V_{M,\rho \otimes \sigma _m}=\Gamma \backslash (V_\rho \otimes V_{\sigma _m})\) over M. Denoting \(E\rightarrow M\) the bundle over M obtained from \(V_\rho \) by quotienting by \(\Gamma \) and \(S^m_0T^*M\) the bundle of trace-free symmetric tensors of order m on M, the bundle \(V_{M,\rho \otimes \sigma _m}\) is isomorphic to the bundle \(\mathcal {E}:=(S^m_0T^*M\oplus S^{m-1}_0T^*M)\otimes E\). There is a differential operator \(A^2_{\sigma _m}\) on \(\mathcal {E}\) constructed from the Casimir operator that has eigenvalues in correspondence with the zeros/poles of \(Z_{S,\sigma _m}(s)\), it is given \(A^2_{\sigma _m}=-\Omega -c(\sigma _m)\) where \(\Omega \) is the Casimir operator and \(c(\sigma )=n^2/4-|\mu (\sigma _m)|^2-2\mu (\sigma ).\rho _{\mathrm{so(n)}}\) with \(\mu (\sigma _m)\) the highest weight of \(\sigma \) and \(\rho _\mathrm{so(n)}=(\tfrac{n}{2}-1,\tfrac{n}{2}-2,\dots ,0)\). Here we have \(\mu (\sigma _m)=(m,0,\dots ,0)\) thus

$$\begin{aligned} c(\sigma _m)=\frac{n^2}{4}-m(m+n-2). \end{aligned}$$

We then obtain the formula

$$\begin{aligned} A^2_{\sigma _m}=(\Delta _m-c(\sigma _m))\oplus ( \Delta _{m-1}-c(\sigma _m)) \end{aligned}$$

where \(\Delta _m=\nabla ^*\nabla -m(m+n-1)\) is the Lichnerowicz Laplacian on (twisted) trace-free symmetric tensors of order m on M (see for instance [38, Section 5]). Now we have by [38, Lemma 5.2] that \(D^*\Delta _m=\Delta _{m-1}D^*\) if \(D^*\) is the divergence operator defined by \(D^*u=-{{{\,\mathrm{Tr}\,}}}(\nabla u)\), and whose adjoint is \(D=\mathcal {S}\nabla \) is the symmetrised covariant derivative. This gives \(\Delta _mD=D\Delta _{m-1}\), but since D is elliptic with no kernel by [40, Proposition 6.6], it has closed range and D gives an isomorphism

$$\begin{aligned} D: \ker (\Delta _{m-1}-c(\sigma _m)-s)\rightarrow \ker (\Delta _{m}-c(\sigma _m)-s)\cap (\ker D^*)^\perp \end{aligned}$$

for each \(s\in {\mathbb {R}}\). In particular, one obtains that for each \(s\in {\mathbb {R}}\)

$$\begin{aligned}&\dim \ker (\Delta _{m}-c(\sigma _m)-s)-\dim \ker (\Delta _{m-1}-c(\sigma _m)-s)\\&\quad =\dim ( \ker (\Delta _{m}-c(\sigma _m)-s)\cap \ker D^*). \end{aligned}$$

Now by [6, Theorem 3.15], the function \(Z_{S,\sigma _m}(s)\) has a zero at s of order

$$\begin{aligned} \begin{aligned}&2 \dim (\dim ( \ker (\Delta _{m}-c(\sigma _m)\cap \ker D^*)) \text { if }s=\tfrac{n}{2}\\&\dim (\dim ( \ker (\Delta _{m}-c(\sigma _m)\cap \ker D^*)) \text { if }s\not =\tfrac{n}{2}. \end{aligned} \end{aligned}$$

\(\square \)

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Dang, N.V., Guillarmou, C., Rivière, G. et al. The Fried conjecture in small dimensions. Invent. math. 220, 525–579 (2020). https://doi.org/10.1007/s00222-019-00935-9

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