Torsion (continued from previous lecture)

Locally homogeneous setting

We now consider a compact locally symmetric manifold \( M = \Gamma \backslash X \), with a local system \( \mathscr L \) coming from a representation \( \rho: G \to \mathrm{SL}(V) \) of the whole Lie group (we need \( \Gamma \) to stabilise a lattice in \( V \)).

To compute the analytic torsion it is also convenient to use the heat kernel. For this recall the trace formula for the heat kernel:
\[
\mathrm{tr}(e^{-t\Delta_q}) := \sum_{j \ge 0} e^{-t\lambda_j} = \int_M \mathrm{tr}(e^{-t\Delta_q}(x, x)) dx
\]
where \( \lambda_j \) are the eigenvalues of \( \Delta_q \) in \( L^2C^q(M) \). With this equality, we can use the Mellin transform to get the following expression for the spectral zeta function \( \zeta_q \) of \( \Delta_q \):
\[
\zeta_q(s) = \frac 1 {\Gamma(s)} \int_0^{+\infty} t^s \mathrm{tr}(e^{-t\Delta_q}) \frac {dt} t.
\]
This expression can be adapted to the \( L^2 \)-setting, and we define the \( L^2 \)-torsion of \( X \) with coefficients in \( \rho \) to be:
\[
\tau_X^{(2)}(\rho) = \frac 1 2 \frac d{ds} \left( \frac 1 {\Gamma(s)} \int_0^{+\infty} dt t^{s-1} \int_{\widehat G} d_\pi e^{-t(\rho(\Omega) – \pi(\Omega))} d\mu_{pl}(\pi) \right)_{s=0}
\]
where:

• For a unitary representation \( \pi \), we put \( d_\pi = \sum_q q(-1)^q \dim \left( \wedge^q\mathfrak p^* \otimes \mathcal H \otimes \pi \right)^K \);
• \( \Omega \) is the Casimir operator of \( G \);
• \( \mu_{pl} \) is the Plancherel measure of \( G \).

The hope is that one can prove for analytic torsion an apptoximation result similar to that for Betti numbers, namely that:
\[
\frac{\tau_M(\mathscr L)}{\mathrm{vol}(M)} \overset{M \to X}{\to} \tau_X^{(2)}(\rho)
\]
where \( M \to X \) can mean that \( \mathrm{inj}(M) \to +\infty \).

For applications to the growth of torsion homology via the Cheeger–Müller theorem one needs in addition to control the regulators \( \overline R^q \), which might be feasible only in an arithmetic setting where additional features such as Hecke operators are available.

Conjecture (Bergeron–Venkatesh): Let \( G = \mathrm{SL}_2({\mathbb C}) \), let \( \Gamma \) be an arithmetic lattice in \( G \) and \( \Gamma_n \) a sequence of congruence subgroups in \( \Gamma \). Then:
\[
\lim_{n\to+\infty} \frac{\log|H_1(\Gamma_n; {\mathbb Z})_{\mathrm{tors}}|}{\mathrm{vol}(M_n)} = -\tau_{{\mathbb H}^3}^{(2)}(1) = \frac 1 {6\pi}
\]

Remarks:

• This conjecture is very well supported by existing computations up to relatively high level;
• There are partial results towards controlling the regulators in congruence covers of arithmetic manifolds (Bergeron–Şengün–Venkatesh);
• In the nonarithmetic case it is likely that the regulators contributes to the putative exponential growth of analytic torsion; on the other hand, in a sequence with \( H^1\otimes{\mathbb Q} = 0 \) the convergence might actually be faster than in a similar arithmetic sequence (this is coherent with probabilistic models for the distribution of small eigenvalues).

It is also possible to prove results similar to the conjectures above, in the case of certain non-trivial local systems. As an example let \( \alpha_{can} \) be the tautological representation \( \mathrm{SL}_2({\mathbb C}) \to \mathrm{SL}_2({\mathbb C}) \), then for any lattice \( \Gamma \subset \mathrm{SL}_2({\mathbb C}) \) we get a local system \( \mathscr L_{\\alpha_{can}} \) (of complex vector spaces, but on some similar systems there can be put an integral structure) which is acyclic in the case where \( M = \Gamma \backslash {\mathbb H}^3 \) is compact. Thus it defines a Reidemeister torsion \( \tau(M, \alpha_{can}) \), and the Cheeger–Müller theorem applies and shows that \( \tau(M, \alpha_{can}) = \tau_M(\mathscr L_{\alpha_{can}}) \).

It turns out that for representation-theoretical reasons, there is a uniform spectral gap for all Laplace operators \( \Delta_q \) on forms with coefficients in \( \mathscr L_{\alpha_{can}} \). This allows to prove the following result.

Theorem (Bergeron–Venkatesh): Let \( \Gamma_n \) be a sequence of torsion-free, cocompact lattices in \( \mathrm{SL}_2({\mathbb C}) \). Let \( M_n = \Gamma_n \backslash {\mathbb H}^3 \) and suppose that \( \mathrm{inj}(M_n) \to +\infty \), then:
\[
\lim_{n \to +\infty} \frac{\log \tau(M_n, \alpha_{can})}{\mathrm{vol}(M_n)} = \frac {11} {12\pi}.
\]

Computations

Now we will explain how to compute the \( L^2 \)-invariants
\[
\beta_q^{(2)}(X) = \lim_{t\to+\infty} \left( \int_{\widehat G} e^{t\pi(\Omega)} \dim \left( \wedge^q \mathfrak p^* \otimes \mathcal H_\pi \right)^K d\mu_{pl}(\pi) \right)
\]
and
\[
\tau_X^{(2)}(\rho) = \frac 1 2 \frac d{ds} \left( \frac 1 {\Gamma(s)} \int_0^{+\infty} dt t^{s-1} \int_{\widehat G} d_\pi e^{-t(\rho(\Omega) – \pi(\Omega))} d\mu_{pl}(\pi) \right)_{s=0}.
\]
In particular this involves describing the Plancherel measure \( \mu_{pl} \) and computing the Casimir eigenvalues \( \pi(\Omega) \).

Harish-Chandra’s Plancherel theorem

Call an irreducible unitary representation \( \pi, \mathcal H_\pi \) of \( G \) a discrete series if it embeds into the (left) regular representation on \( L^2(G) \). Denote by \( \widehat G_d \subset \widehat G \) the subset of discrete series representations

Theorem (Harish-Chandra): Let \( K \) be a maximal compact subgroup of \( G \). Then \( G \) has discrete series if and only if \( \mathrm{rank}_{\mathbb C}(G) = \mathrm{rank}_{\mathbb C}(K) \).

The quantity \( \delta(G) := \mathrm{rank}_{\mathbb C}(G) – \mathrm{rank}_{\mathbb C}(K) \) is often called the fundamental rank of \( G \). For \( G = \mathrm{SL}_m({\mathbb R}) \) it is equar to \( m – 1 – \lceil m/2 \rceil \) so that \( G \) has discrete series if and only if \( m=2 \). For \( G = \mathrm{SO}(m, 1) \) it is equal to 0 if m is even and 1 if it is odd.

One way to construct discrete series for \( G = \mathrm{SL}_2({\mathbb R}) \) is as follows. Let \( \omega \) be a differential harmonic 1-form on the hyperbolic plane \( {\mathbb H}^2 \). It is possible to solve explicitely the differential equation \( d\omega = 0 \) on \( {\mathbb H}^2 \), and then there is a 1-dimensional subspace in the solutions which is composed of square-integrable forms. Taking \( \phi \) to be a vector in \( \mathrm{Hom}_K(\mathfrak p, C^\infty(G) ) \) corresponding to one of these, the subspace obtained as the closure of the linear span of the orbit \( G \cdot \phi \) is a discrete series for \( G \).

In general unitary representations are constructed by inducing discrete series from the Levi components of parabolic subgroups. We will describe this construction in the case of \( G = \mathrm{SL}_m({\mathbb R}) \). Let \( K = \mathrm{SO}(m) \) and let \( T_f \) (« fundamental torus ») be the maximal torus in \( K \) obtained as the subgroup:
\[
T_f = \left( \begin{array}{cccc}\mathrm{SO}(2) & 0 & \cdots & 0 \\
0 & \mathrm{SO}(2) & \ddots & \vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & \mathrm{SO}(2) \end{array}\right) \text{ or }
\left( \begin{array}{cccc}\mathrm{SO}(2) & 0 & \cdots & 0 \\
0 & \ddots & \ddots & \vdots\\
\vdots & \ddots & \mathrm{SO}(2) & 0 \\
0 & \cdots & 0 &1\end{array}\right)
\]
depending on whether \( m \) is even or odd. Let \( \mathfrak b = \mathfrak t_f \otimes {\mathbb C} \) and \( \mathfrak t_u = \mathfrak b \oplus \mathfrak a_0 \) a Cartan subalgebra containing \( \mathfrak t_f \), so that \( \mathfrak a_0 \) is the \( {\mathbb C} \)-span of the Lie algebra of a maximal \( {\mathbb R} \)-split torus \( A_f \). Explicitely we have
\[
A_f = \left(\begin{array}{ccccc}
\lambda_1 & 0 & \cdots & \cdots & 0 \\
0 & \lambda_1 & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \lambda_{m/2} & 0\\
0 & \cdots & \cdots &0&\lambda_{m/2}\end{array}\right)
\]
in case m is even and similar otherwise. Finally, let
\[
M_f = \left( \begin{array}{cccc}\mathrm{SL}_2({\mathbb R}) & 0 & \cdots & 0 \\
0 & \mathrm{SL}_2({\mathbb R}) & \ddots & \vdots \\
\vdots & \ddots & \ddots & 0 \\
0 & \cdots & 0 & \mathrm{SL}_2({\mathbb R}) \end{array}\right)
\]
if m is m is even and similar otherwise. Then the parabolic subgroup \( P_f = M_f A_f N \) is called the « fundemantal parabolic subgroup »; it is maximal among parabolic subgroups of \( G \) whose Levi component (here it is \( M_f \cong \mathrm{SL}_2({\mathbb R})^{\lceil m/2 \rceil} \))has discrete series. Now let \( Q \) be any parabolic subgroup contained in \( P_f \) and write \( Q = M_Q A_Q N_Q \) so that \( M_Q \cong \mathrm{SL}_2({\mathbb R})^r\) for some \( 0 \le r \le \lceil m/2 \rceil \) has discrete series.

Let \( \theta \) be a discrete series of \( M_Q \), and \( \nu \in \mathfrak a_Q^* \). We can construct a representation \( \pi(\theta, \nu) \) of \( G \) by inducing from \( Q \) the representation \( \theta \otimes \exp(\nu) \otimes 1 \) of \( Q \). This acts on the Hilbert space:
\[
\mathcal H_{\pi(\theta, \nu)} = \{ f: G \to \mathcal H_\theta : f(gman) = \theta(m) \cdot e^{-(\nu + \rho_Q)(\log a)} f(g), f|_{Q\backslash G} \in L^2 \}
\]
where \( \rho_Q \) is a normalising factor equal to the half-sum of positive roots of \( \mathfrak a_Q \). For \( \nu \in i\mathfrak a_{Q, {\mathbb R}}^* \) the representation \( \pi(\theta, \nu) \) is unitary, and the Casimir eigenvalue is given by:
\[
\pi(\theta, \nu)(\Omega) = \langle \nu, \nu \rangle – \langle \rho_Q, \rho_Q \rangle + \theta(\Omega_M).
\]

Plancherel formula : Let \( f \in \mathscr C(G) \). Then:
\[
f(e) = \sum_{Q \subset P_f} \sum_{\theta \in (\widehat{M_Q})_d} \int_{a_{Q,{\mathbb R}}^*} \mathrm{tr}(\pi(\theta, i\nu)(f)) p_\theta(\nu) d\nu
\]
where \( p_\theta \) are explicit polynomials.

In general the representations are obtained in a similar manner. Applying this to the heat kernel, we obtain:
\[
\mathrm{tr}(k_t(e)) = \sum_Q \sum_\theta \int_{a_{Q,{\mathbb R}}^*} e^{-t\|\nu\|^2 + \|\rho_Q\|^2 – \theta(\Omega_M)} \dim(\pi(\mathcal H_{\pi(\theta, i\nu)} \otimes \theta, i\nu))^K p_\theta(\nu) d\nu.
\]
As \( t \to +\infty \), all integrals for \( Q \not= G \) tend to 0 and thus we see that if \( P_f \not= G \), that is if \( \delta(G) > 0 \), then \( \beta_q^{(2)}(X) = 0 \) for all \( Q \). Similarly one can see that the \( L^2 \)-torsion \( \tau_X^{(2)}(\rho) \) vanishes if \( \delta(G) \not= 1 \) (independently of the representation \( \rho \)).