Category Archives: Hausdorff Institute

L2-invariants of locally symmetric spaces, III (Nicolas Bergeron)

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 \( G \).

Continue reading →

L2-invariants of locally symmetric spaces, II (Nicolas Bergeron)

Betti numbers of symmetric spaces (continuing previous lecture)

Recall the setup: \( G, K, X=G/K \) are respectively a semisimple Lie group, a maximal compact subgroup and the associated symmetric space; \( \Gamma, M = \Gamma \backslash S \) are a discrete, torsion-free subgroup and the associated \( X \)-manifold. The de Rham complex of \( M \) can be computed via:
\[
C^q(M) = \mathrm{Hom}_K \left( \wedge^q\mathfrak p, C^\infty(\Gamma \backslash G) \right)
\]
and there is a Laplace operator \( \Delta_q \) which is essentially auto-adjoint and positive on the \( L^2 \)-completion of \( C^q \), and an associated heat operator which is given by convolution with the heat kernel
\[
e^{-t\Delta_q} \in \left( \mathrm{End}(\wedge^q \mathfrak p^*) \otimes C^\infty(\Gamma \backslash G \times \Gamma \backslash G) \right)^{K \times K}.
\]
The Betti numbers of \( M \) are then given by the formula:
\[
b_q(M) = \lim_{t \to +\infty} \int_M \mathrm{tr}(e^{-t\Delta_q}(x, x)) dx.
\]

Continue reading →

L2-invariants of locally symmetric spaces, I (Nicolas Bergeron)

Betti numbers of locally symmetric spaces

Let \( G \) be a semisimple Lie group, for example:

  • \( G = \mathrm{SL}_2({\mathbb R}) \) or \( \mathrm{SL}_m({\mathbb R}), m \ge 3 \);
  • \( G = \mathrm{SO}(p, q), p, q \ge 1 \) ;
  • \( G = \mathrm{SL}_2({\mathbb C}) \), which is isogenic to \( \mathrm{SO}(3,1) \).

Let \( K \) be a maximal compact subgroup of \( G \), and let \( X = G/K \) be the associated Riemannian symmetric space, whose group of orientation-preserving isometries is \( G \) (at least if the latter is connected). In the examples:

  • If \( G = \mathrm{SL}_2({\mathbb R}) \), we can take \( K = \mathrm{SO}(2) \) and \( X \) is then the hyperbolic plane \( {\mathbb H}^2 \);
  • If \( G = \mathrm{SL}_2({\mathbb C}) \), \( K = \mathrm{SU}(2) \) then \( X \) is the hyperbolic space \( {\mathbb H}^3 \);
  • If \( G = \mathrm{SL}_m({\mathbb R}) \) and \( K = \mathrm{SO}(m) \) then \( X \) is the subset of the projective space of m by m matrices which is the image of the cone of positive definite matrices.

Continue reading →