A formula for distribution trace characters on nilpotent Lie groups

Abstract:

The distribution trace character ${\theta _\pi }$ of an irreducible representation $\pi$ on a simply connected nilpotent Lie group $N$ is described as a superposition of unitary characters on certain subgroups, in terms of the canonical objects introduced by R. Penney [8]. For $l \in {\mathfrak {n}^*}$, define ${\mathfrak {h}_1}(l) =$ smallest ideal containing the radical $\mathfrak {r}(l)$, and ${\mathfrak {h}_{k + 1}}(l) = {\mathfrak {h}_1}(l|{\mathfrak {h}_k}(l))$. These subalgebras terminate in a subordinate subalgebra ${\mathfrak {h}_\infty }(l)$ after finitely many steps. If ${H_\infty } = \exp ({\mathfrak {h}_\infty })$, ${\mathcal {X}_\infty } = ({e^{2\pi il}}) \circ \log |{H_\infty }$, and $({\mathcal {X}_\infty },{H_\infty }) \cdot n = ({\mathcal {X}_\infty } \cdot n,\;{H_\infty } \cdot n)$, where ${H_\infty } \cdot n = {n^{ - 1}}{H_\infty }n$, ${\mathcal {X}_\infty }\cdot n(h’) = {\mathcal {X}_\infty }(nh’{n^{ - 1}}){\text { on }}{H_\infty } \cdot n$, then $\left \langle {{\theta _\pi },\phi } \right \rangle = {\smallint _{{K_\infty }\backslash N}}\left \langle {({\mathcal {X}_\infty },{H_\infty }) \cdot n,\phi } \right \rangle d\dot n$, where the pair $({\mathcal {X}_\infty },{H_\infty }) \cdot n$ is regarded as the tempered distribution $\left \langle {({\mathcal {X}_\infty },{H_\infty }) \cdot n,\phi } \right \rangle = {\smallint _{{H_\infty }}}{\mathcal {X}_\infty }(h)\phi ({n^{ - 1}}hn)dh$, and where ${\mathfrak {k}_\infty } = \{ X \in \mathfrak {n}:l[{h_\infty },X] = 0\}$ gives the stabilizer of the pair $({\mathcal {X}_\infty },{H_\infty })$. The integral over ${K_\infty }\backslash N$ is absolutely convergent for any Schwartz function $\phi$ on $N$.