TY - JOUR

T1 - An Index Formula for Groups of Isometric Linear Canonical Transformations

AU - Savin, Anton

AU - Schrohe, Elmar

N1 - Funding Information: We thank Gennadi Kasparov for pointing out the Bott periodicity theorem in [19] to us. The work of the first author was supported by the Ministry of Science and Higher Education of the Russian Federation: agreement no. 075-03-2020-223/3 (FSSF-2020-0018); that of the second by DFG through project SCHR 319/8-1.

PY - 2022

Y1 - 2022

N2 - We define a representation of the unitary group \(U(n)\) by metaplectic operators acting on \(L^2(\mathbb{R}^n)\) and consider the operator algebra generated by the operators of the representation and pseudodifferential operators of Shubin class. Under suitable conditions, we prove the Fredholm property for elements in this algebra and obtain an index formula.

AB - We define a representation of the unitary group \(U(n)\) by metaplectic operators acting on \(L^2(\mathbb{R}^n)\) and consider the operator algebra generated by the operators of the representation and pseudodifferential operators of Shubin class. Under suitable conditions, we prove the Fredholm property for elements in this algebra and obtain an index formula.

KW - Index theory

KW - metaplectic operators

KW - Shubin class pseudodifferential operators

UR - http://www.scopus.com/inward/record.url?scp=85134665983&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2008.00734

DO - 10.48550/arXiv.2008.00734

M3 - Article

VL - 27

SP - 983

EP - 1013

JO - Documenta mathematica

JF - Documenta mathematica

SN - 1431-0635

ER -