1
$\begingroup$

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 $\mathrm{Ind}_K^G$?

$\endgroup$
4
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks, I didn't realize that this was in Bushnell-Henniart. Inspecting the proof there, I see that my assumptions were too restrictive: the same conclusion holds even if $K$ is merely closed (instead of being open) and with $\mathbb{C}$ replaced by any commutative ring $R$ such that $G$ has a compact open subgroup with pro-order in $R^*$. $\endgroup$ – Question Mark Oct 24 '14 at 16:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.