About representation theory of Heisenberg group

Question

Actually I am an undergraduate student, but I want to study Heisenberg groups over arbitrary field.

Firstly, why is this group important? I know that the Heisenberg group is important in the field of quantum physics, but in mathematics how is it? And secondly I want to know the current studies of the classifications of the irreducible representations of Heisenberg groups over an arbitrary field(like over $\mathbb{Q}_p,\mathbb{R}, \mathbb{C}$ or finite field). Would you recommend any references in this area? (e.g. survey papers or influential papers)

Thanks.

Answer 1

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

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PGL{PGL}$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 \begin{align*} \Sp(V) & {}\rightarrow \PGL(W) \\ g & {}\mapsto \phi_g. \end{align*}

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: Gan - Representations of metaplectic groups.

Answer 2

This seems like a reasonable survey article: Howe - On the role of the Heisenberg group in harmonic analysis.

Also there is a book by Deitmar and Echterhoff - "Principles of Harmonic Analysis", which I found quite readable, but that treats coefficients in $\mathbb{R}$ only. But this book is probably more likely to be useful for graduate students.

Answer 3

You should really study the representation theory in the wider context of a semidirect product. George Mackey wrote the basic theory on this, and you'll almost certainly understand more by working out what the theory says for these special cases (whether discrete groups or more general locally compact groups).