Theta correspondence

Remarks

The theorem used in the definition is Howe's duality theorem for real groups; the same definition over $p$-adic fields rests on the work of several later authors.

Example: let $W = \R^2$, so $\mathrm{Sp}(W) = \mathrm{SL}_2(\R)$, and let $G = \set{\pm 1}$, which is the orthogonal group $\mathrm{O}(1)$. The centralizer of $\set{\pm 1}$ is all of $\mathrm{SL}_2(\R)$, and the centralizer of $\mathrm{SL}_2(\R)$ is its center $\set{\pm 1}$, so $(\mathrm{O}(1), \mathrm{SL}_2(\R))$ is a dual pair. In the Schrödinger model on $L^2(\R)$, the element $-1$ acts by $\psi(x) \mapsto \psi(-x)$, up to a constant depending on the lift. So the two characters of $\mathrm{O}(1)$ correspond to the two irreducible pieces of the oscillator representation of $\mathrm{Mp}_2(\R)$, on even and on odd functions.

Non-example: in $\mathrm{SL}_2(\R)$, let $A = \set{\operatorname{diag}(a, a^{-1}) : a > 0}$. The centralizer of $A$ is $\set{\operatorname{diag}(a, a^{-1}) : a \ne 0}$, which is larger than $A$, so $(A, A)$ is not a dual pair.

The standard dual pairs include $(\mathrm{O}(p, q), \mathrm{Sp}_{2m}(\R))$ in $\mathrm{Sp}(\R^{p+q} \otimes \R^{2m})$ and $(\mathrm{U}(p, q), \mathrm{U}(r, s))$.