Q:

Does restriction to an open subgroup preserve projective smooth representations?

Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth $\mathbb{C}$-linear representations of $K$ preserve projective objects?
One is tempted to make an argument using the right adjoint $\mathrm{Ind}_K^G$, right exactness of which would suffice. But does the openness of $K$ imply the right exactness of the induction functor $\mathrm{Ind}_K^G$?

A:

The induction functor is actually exact, not just right-exact, so yes, you can do this. Seeing as you've essentially asked two questions about the induction functors on these groups, you might want to take a look at Bushnell--Henniart's Local Langlands for GL2; section 2 of this gives a pretty simple introduction to the two functors and their basic properties.

