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Abstract LetGbe a linear real reductive Lie group. Orbital integrals define traces on the group algebra ofG. We introduce a construction of higher orbital integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra ofG. We analyze these higher orbital integrals via Fourier transform by expressing them as integrals on the tempered dual ofG. We obtain explicit formulas for the pairing between the higher orbital integrals and theK-theory of the reduced group$$C^{*}$$-algebra, and we discuss their application toK-theory.
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Local Rankin–Selberg integrals for Speh representations
We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a $$p$$ -adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model.
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- Award ID(s):
- 1700637
- PAR ID:
- 10172643
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 156
- Issue:
- 5
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 908 to 945
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X2000706X
