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!

allvectors in the Hilbert space... it does exist on thesmoothvectors. 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