Representation Theory: Basics and Heisenberg Representation (3) May 25, 2010

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We return to our regularly scheduled programming — more notes from the seminar.

The next example is the Heisenberg group. Recall that this is the group of 3-by-3 upper triangular matrices with ones on the diagonal. What are the irreducible representations of this group H?

Let $(V, \omega) = (\mathbb{F}_p \times \mathbb{F}_p; \omega(v, v') = det(v v')$. V is denoted a symplectic vector space, and $\omega$ is its form. The Heisenberg group H is
$H = V \times \mathbb{F}_p$
endowed with the group law
$(v, z) * (v', z') = (v + v', z + z' + 1/2 \omega (v, v'))$

One-dimensional representation:
$H \to V \to \mathbb{C}^\times$
$\rho_\theta(h) = \theta(Pr(h))$
where $\theta$ is the function from $V to \mathbb{C}^\times$. So this is just the projection.

Stone-von Neumann Theorem:

There exists a unique up to isomorphism irreducible representation $latex(\pi, H, \mathcal{H})$ such that
$\pi |_z = \psi(z) Id|_{\mathcal{H}}$

Lagrangian models.

Space:
$\mathcal{H}_L = \mathbb{C}(L \ H, \psi) = \{ f: H \to \mathbb{C}, f(z l h) = \psi(z) f(h)$
$dim \mathcal{H}_L = p$
We can think of L as a line through the space V.

Action
$\pi_L = H \to \mathcal{H}_L$
$[\pi_L(h') f](h) = f(h h')$

Claim $(\pi_L, H, \mathcal{H})$ irreducible.
We obtain $(p-1) \times p$-dimensional irreducible representations
$P^2 * 1^2 + (p-1)*p^2 = p^3 = |H|$

Intertwiners:
$F_{M, L} \in Hom_H(\mathcal{H}_L, \mathcal{H}_M) \simeq \mathbb{C}$
$F_{M, L}[f] (h) = \sum_{m \in M} f(mh)$