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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A formula for distribution trace characters on nilpotent Lie groups

Authors: L. Corwin and F. P. Greenleaf
Journal: Proc. Amer. Math. Soc. 89 (1983), 738-742
MSC: Primary 22E27; Secondary 22E25
MathSciNet review: 719008
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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\vert{\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 \vert{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 }\b... ...angle {({\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$.

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