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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A formula for distribution trace characters on nilpotent Lie groups
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by L. Corwin and F. P. Greenleaf PDF
Proc. Amer. Math. Soc. 89 (1983), 738-742 Request permission

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 $N$.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 89 (1983), 738-742
  • MSC: Primary 22E27; Secondary 22E25
  • DOI: https://doi.org/10.1090/S0002-9939-1983-0719008-1
  • MathSciNet review: 719008