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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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ISSN
1435-5345
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Volume 2018, Issue 745

# Addendum to “Singular equivariant asymptotics and Weyl’s law”

Pablo Ramacher
• Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Str., 35032 Marburg, Germany
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Published Online: 2017-02-25 | DOI: https://doi.org/10.1515/crelle-2017-0001

## Abstract

Let M be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group G. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on ${T}^{\ast }M×G$ with singular critical sets that were examined in [7] in order to determine the asymptotic distribution of eigenvalues of an invariant elliptic operator on M. As an immediate consequence, we deduce from this an asymptotic multiplicity formula for families of irreducible representations in ${\mathrm{L}}^{2}\left(M\right)$. The improved remainder is used in [4] to prove an equivariant semiclassical Weyl law and a corresponding equivariant quantum ergodicity theorem.

## References

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H. Donnelly, G-spaces, the asymptotic splitting of ${L}^{2}\left(M\right)$ into irreducibles, Math. Ann. 237 (1978), 23–40. Google Scholar

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A. Grigis and J. Sjöstrand, Microlocal analysis for differential operators, London Math. Soc. Lecture Note Ser. 196, Cambridge University Press, Cambridge 1994. Google Scholar

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B. Küster and P. Ramacher, Quantum ergodicity and symmetry reduction, J. Funct. Anal. (2017), to appear.

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O. Paniagua-Taobada and P. Ramacher, Equivariant heat asymptotics on spaces of automorphic forms, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3509–3537. Google Scholar

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P. Ramacher, Singular equivariant asymptotics and the momentum map. Residue formulae in equivariant cohomology, J. Symplectic Geom. 14 (2016), no. 2, 449–539.

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P. Ramacher, Singular equivariant asymptotics and Weyl’s law. On the distribution of eigenvalues of an invariant elliptic operator, J. reine angew. Math. 716 (2016), 29–101.

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V. S. Varadarajan, Lie groups, Lie algebras and their representations, Englewood Cliffs, Prentice Hall 1974. Google Scholar

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N. R. Wallach, Real reductive groups. Vol. I, Academic Press, New York 1988. Google Scholar

Published Online: 2017-02-25

Published in Print: 2018-12-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 745, Pages 281–293, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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