Definition 4

Let G be a topological group, and let $\varrho$ and $\pi$ be unitary representations of G in Hilbert spaces E and F respectively. A G-invariant sesquilinear form on $E\times F$ is a continuous sesquilinear form $q$ on $E\times F$ such that

for every $g\in G$ and every $(x, y)\in E\times F$.

The vector space of G-invariant sesquilinear forms on $E\times F$ is denoted by Sesq$_G(\varrho , \pi )$ or Sesq$_G(E,F)$.