Let $G$ be a nilpotent Lie Group, and $\pi:G\to B(\mathcal H)$ be an irreducible unitary representation on the Hilbert space $\mathcal H$. One can use the Bochner integral to define a linear map as follows: $$\hat\pi:L^1(G)\to B(\mathcal H),\;\hat\pi(\phi)(v)=\int_G\phi(g)\pi(g)(v)dg\text{ for any }v\in \mathcal H.$$

My first question is: is there any study of $\hat\pi(\phi)$ in the case when $\phi$ is a generalized function, say $\phi=\sum_i \delta_{g_i}$? Here $\delta_{g_i}$ is the dirac delta function on $G$.

My second question is more involved: Assume $\mathcal C=\{g_i|1\le i\le 2k\}$ with $\prod_{i=1}^{2k}g_i=e$, then let $\phi_{\mathcal C}=\sum_{i=1}^k\left(\delta_{\prod_{j=1}^{2i}g_{j}}-\delta_{\prod_{j=1}^{2i+1}g_j}\right)$, is it true that the sum of $\hat\pi(\phi_{\mathcal C})\mathcal H$ is dense in $\mathcal H$, or dense in some subspace of finite codimension? Here, the sum is taken over all nontrivial $\mathcal C=\{g_i|1\le i\le 2k\}$ for all $k\ge 2$.

I am trying to seek for an answer to the above question, but any reference is also appreciated!

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    $\begingroup$ A Dirac delta at $g$ would just act by $\pi(g)$. Its derivative would act by the derivative of $\pi(g)$, which doesn't exist in application to all vectors in the Hilbert space... it does exist on the smooth vectors. As suggested already by this, compactly supported distributions on $G$ do have an action on smooth vectors, since such distributions are derivatives of continuous functions. Your second question seems to me more convoluted than it really is, namely, to my perception, you are asking whether the sum of translates by $g$'s is dense, which is is for irreducibles. $\endgroup$ – paul garrett Nov 3 '16 at 22:02
  • $\begingroup$ Thanks for your reply and remarks. I am not quite clear about your last sentence, could you please explain a little bit more that why it is for irreducible. Thanks again. $\endgroup$ – Changguang Nov 3 '16 at 22:12
  • $\begingroup$ The closure of the collection of all finite linear combinations of vectors $\pi(g)(v)$ is a (non-zero) $G$-stable topologically closed subspace of the Hilbert space $H$. If $H$ is an irreducible repn (topological irreducibility is "understood"!), then such a closed, $G$-stable subspace can only be the whole thing. Is that addressing your question properly? $\endgroup$ – paul garrett Nov 3 '16 at 22:14
  • $\begingroup$ Yes, this certainly clarifies my question. Thank you. However, I don't see how this helps for my second question! $\endgroup$ – Changguang Nov 3 '16 at 22:21

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