Deequivariantisation of indecomposable sheaves Let G be a connected group acting on a space X. All spaces should be reasonable, so e.g. G is a complex affine algebraic group acting on an algebraic variety X, with everything done using the usual complex topology.
Then for any coefficient field k, there is an equivariant derived category $D^b_G(X;k)$, together with a forgetful functor to $D^b(X;k)$.
Can you give an example of an indecomposable object in $D^b_G(X;k)$ which becomes decomposable in $D^b(X;k)$?
 A: Take $X=pt$ and $G=G_m$ and $k$ to have characteristic zero. The equivariant derived category in this case is equivalent to modules for the homology of the circle, ie exterior algebra on a generator in degree -1. The augmentation module (trivial representation of $G$) has self-Exts given by $H^*(BG_m)=k[u]$, polynomials on a variable of degree 2. So one can make interesting indecomposable objects as self-extensions of the trivial rep and they all become decomposable when we forget the $G$ action (ie pull back to a point) -- for example there's a canonical extension of $k$ by $k[1]$ corresponding to $u$ itself. This is just the pushforward of the constant sheaf from $X$ to $BG_m$, aka the "regular representation", action of $G_m$ on its own cohomology.
Edit: Some elaboration in response to comments.
Let $G$ be any group (topological or algebraic, depending on what setting you're in), and let's stick to char. 0 to be safe. The (dg-enhanced) equivariant derived category of a point is equivalent to modules for the dg algebra $C_*(G)$ of chains on $G$ under convolution (the ``topological group algebra"). Let's call this category topological representations of $G$ (in the de Rham setting of D-modules we can equivalently speak of algebraic reps of $G$ with a trivialization of the Lie algebra action).
Pullback to a point is the forgetful functor from topological reps of $G$ to chain complexes.
This category contains the trivial representation = augmentation module = constant sheaf on $BG$. Its Ext algebra is $C^*(BG)$, cochains on $BG$ (or $H^*_G(pt)$ on level of cohomology).
For $G$ connected as in the question, the trivial module is a ``generator" in a weak sense (it's not compact). This is expressed eg by Koszul duality, given by Ext from the trivial module, relating $C_*(G)-mod$ and $C^*(BG)-mod$ -- an equivalence with suitable boundedness or after suitable completions. In any case we can produce representations by iterated extensions (cones of self maps) of the augmentation (again being careful about noncompactness/boundedness issues).
So to find examples as in the question we just need to find indecomposable $C_*(G)$-modules of dim > 1, since all such are certainly decomposable in $Vect_k$. Taking a nonzero self-Ext of the trivial module -- aka nonzero cohomology class of $BG$ -- and taking its cone we get examples.
A: Take $G_m$ acting on itself via $z\mapsto z^2$, and take $k$ to be of characteristic $2$. The equivariant derived category here is the derived category of $\mathbb{Z}/2\mathbb{Z}$-modules. Now take the regular representation. It’s indecomposable. If you forget the equivariant structure it decomposes.
Note: this gives you something in the heart of your favorite t-structures that satisfies the requirements, or if you like you can use Yoneda Ext to produce a genuine complex in $D^b_{G_m}(G_m) = D^b_{\mathbb{Z}/2\mathbb{Z}}(pt)$ satisfying your requirements. Doesn’t contradict my remark about the forgetful functor being faithful on t-structures (since splitting has to do with existence of idempotents, rather than morphisms becoming $0$).
A: This is essentially a "down to earth" version of what David Ben-Zvi is saying. The object/example produced below essentially matches with what David is suggesting. I am just producing it "geometrically" to convince anyone interested that it is not a particularly strange beast and one doesn't need to know about dg-models or Koszul duality.
Let $\mathbb{G}_m$ act on itself by left multiplication. Let $a\colon \mathbb{G}_m\to pt$ be the canonical map. Consider the "silly" diagram (the functor $a_*$ is derived):
$\require{AMScd}$
\begin{CD}
D^b_{\mathbb{G}_m}(\mathbb{G}_m) @>{For}>> D^b(\mathbb{G}_m)\\
@VV{a_*}V @VV{a_*}V\\
D^b_{\mathbb{G}_m}(pt) @>{For}>> D^b(pt)
\end{CD}
This diagram commutes up to canonical isomorphism (essentially by construction/definition of the equivariant derived category). Write $\underline{k}$ for the (equivariant) constant sheaf on $\mathbb{G}_m$ (i.e., monoidal unit). Clearly,
$$ For(a_*\underline{k})= a_*For(\underline{k}) = H^*(\mathbb{G}_m)= k\oplus k[-1] $$
However, $a_*\underline{k}$ is indecomposable, because
$$Hom(k, a_*\underline{k}) = H^*_{\mathbb{G}_m}(\mathbb{G}_m) = k$$
Here, the first "$k$" on the left hand side is the constant sheaf in $D^b_{\mathbb{G}_m}(pt)$ (this object generates the whole triangulated category, so it will detect decompositions), and as before the "$k$" on the far right hand side is just the field of coefficients.
The point, from the Koszul dual/algebraic perspective, is that any situation with torsion (i.e., where you can't obtain ordinary cohomology from equivariant cohomology) will lead to this. The characteristic 2 example I gave earlier is also "torsion phenomenon" but in a different sense.
