Is torus a symmetric space?

Question

Let $\Lambda$ be a lattice in $\mathbf{R}^{n}$. The flat torus $T:= \mathbf{R}^{n}/\Lambda$ is a homogeneous space. I have two questions:

1. Is $T$ a symmetric space?

2. If so, is it irreducible for any $\Lambda$?

If the explanation is too long to write, just citing the reference will nevertheless be very helpful.

Thank you in advance.

Edit: For 1. I meant global symmetric space.
For 2. If $T$ is a global symmetric space, is the isotropy representation irreducible for every $\Lambda$?

Answer 1

One definition of a symmetric space is the existence of a (local) involution preserving the metric. Since the lattice $\Lambda$ is invariant under the isometry $\lambda \mapsto -\lambda$, so is the torus (even globally).

If by "irreducible" you mean "indecomposable as a metric product", then the torus will be reducible if and only if the lattice is a "rectangular" lattice, i.e., spanned by a pair of orthogonal vectors.