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

*Posted by Sarah in Uncategorized.*

Tags: representation theory

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Tags: representation theory

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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 is denoted a symplectic vector space, and is its form. The Heisenberg group H is

endowed with the group law

One-dimensional representation:

where is the function from . 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

Lagrangian models.

Space:

$dim \mathcal{H}_L = p$

We can think of L as a line through the space V.

Action

Claim irreducible.

We obtain -dimensional irreducible representations

Intertwiners:

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