Here is one important way in which the Heisenberg group is important :

Let $F$ be a local field of characteristic not equal to 2 (so for example, one of your fields $\mathbb{Q}_p$ or $\mathbb{R}$ or $\mathbb{C}$ above), and let $\psi$ be a nontrivial unitary additive character of $F$. Let $V$ be a symplectic space over $F$ with symplectic form $\langle , \rangle$, and form the Heisenberg group $H(V)$. This is the group which is set theoretically $V \times F$, with group operation $(v_1, x_1)(v_2,x_2) = (v_1+v_2, t_1+t_2 + \frac{1}{2} \langle v_1,v_2 \rangle)$.

By the Stone-Von-Neumann theorem, $H(V)$ has a unique irreducible smooth (or unitary) representation $(\rho_{\psi}, W)$ over $\mathbb{C}$ with central character $\psi$. Note that $Sp(V)$ acts on $H(V)$ in a natural way through its action on $V$. Then, by Schur's Lemma, for every $g \in Sp(V)$, there exists an intertwining operator $\phi_g$ between $(\rho_{\psi}, W)$ and $(\rho_{\psi} \circ g, W)$, which is unique up to multiplication by $\mathbb{C}^*$.

We therefore get a projective representation
$$Sp(V) \rightarrow PGL(W)$$
$$g \mapsto \phi_g$$

It can be shown that this projective representation lifts to a representation of $\widetilde{Sp(V)}$, where $\widetilde{Sp(V)}$ is a naturally defined double cover of $Sp(V)$, called the Metaplectic group. This representation, denoted $\omega_{\psi}$, is called the Weil representation. Both the Weil representation and the metaplectic group are important in number theory and representation theory. If you're curious, Wee Teck Gan has a nice survey article on how the metaplectic group arises in the Langlands program : http://www.math.ucsd.edu/~wgan/ICCM.pdf

Best,

Moshe

youshould think the group important if you want to study its irreps. So let me turn the tables and ask:whydoyouwant to study this group? – Theo Johnson-Freyd Apr 8 '11 at 14:36