The definition I know of for a cuspidal automorphic representation of, say, $G=\mathrm{GL}_2$ over a number field $F$ (relative to a choice of compact open subgroup $K_f$ of $G(\mathbf{A}_F^\infty)$ and a maximal compact subgroup $K_\infty$ of $G(F\otimes_\mathbf{Q}\mathbf{R})$) is: an irreducible subquotient (equivalently subspace) of the space of cuspidal automorphic forms for $G$ over $F$. Since the full group $G(\mathbf{A}_F)$ does not preserve the property of $K=K_fK_\infty$-finiteness, "subrepresentation" here really means a subspace which is simultaneously a $G(\mathbf{A}_F^\infty)$-submodule and a $(\mathfrak{g}_\infty,K_\infty)$-submodule (and this can be phrased more concisely as a submodule for the appropriate global Hecke algebra). Now, I gather that there is a more ``analytic" theory, where one actually has a representation of $G(\mathbf{A}_F)$ on a Hilbert space. My first question is:

**Are these theories equivalent in some sense?**

I should make this more precise. I know that any cuspidal automorphic representation in the first sense is unitarizable. If I were to take the completion, would I get a unitary (admissible) Hilbert space representation of $G(\mathbf{A}^\infty)$ whose space of $K$-finite vectors was isomorphic to the representation with which I started?

I also know that two irreducible unitary admissible representations of, say, $\mathrm{GL}_2(\mathbf{R})$, are unitarily equivalent if and only if they are infinitesimally equivalent in the sense of having isomorphic associated $(\mathfrak{gl}_2(\mathbf{R}),\mathrm{O}_2(\mathbf{R}))$-modules (the spaces of $K$-finite vectors). My second question is:

**Does this extend to unitary cuspidal automorphic representations (which I guess would be defined as irreducible admissible subrepresentations of the appropriately defined space of cusp forms in $L^2(G(F)\setminus G(\mathbf{A}_F),\omega)$, $\omega$ being the unitary central character)?**

That is, are two unitary cuspidal automorphic representations isomorphic if and only if their spaces of $K$-finite vectors are isomorphic as modules for the Hecke algebras? And, out of curiosity:

**If the answer to my second question is 'yes,' is the same true with $G$ an arbitrary connected reductive group of a number field?**

The reason I ask is because, in some of the literature, it seems like both viewpoints are being used, sometimes at the same time, sometimes implicitly. So I would like to know if I can freely pass back and forth between them.

Thanks!