TY - JOUR

T1 - Local index formulae on noncommutative orbifolds and equivariant zeta functions for the affine metaplectic group

AU - Savin, Anton

AU - Schrohe, Elmar

N1 - Funding Information: Acknowledgement. ES thanks Gerd Grubb and Jens Kaad for helpful remarks. The work of the first author was supported by RFBR , project number 21-51-12006 ; that of the second by DFG through project SCHR 319/8-1 . We thank the referees for useful suggestions.

PY - 2022/11/19

Y1 - 2022/11/19

N2 - We consider the algebra \(A\) of bounded operators on \(L^2(\mathbb{R}^n)\) generated by quantizations of isometric affine canonical transformations. The algebra \(A\) includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple \((A, H, D)\) with \(H=L^2(\mathbb R^n, \Lambda(\mathbb R^n))\) and the Euler operator \(D\), a first order differential operator of index \(1\). We show that this spectral triple has simple dimension spectrum: For every operator \(B\) in the algebra \(\Psi(A,H,D)\) generated by the Shubin type pseudodifferential operators and the elements of \(A\), the zeta function \({\zeta}_B(z) = {\rm Tr} (B|D|^{-2z})\) has a meromorphic extension to \(\mathbb C\) with at most simple poles. Our main result then is an explicit algebraic expression for the Connes-Moscovici cyclic cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.

AB - We consider the algebra \(A\) of bounded operators on \(L^2(\mathbb{R}^n)\) generated by quantizations of isometric affine canonical transformations. The algebra \(A\) includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple \((A, H, D)\) with \(H=L^2(\mathbb R^n, \Lambda(\mathbb R^n))\) and the Euler operator \(D\), a first order differential operator of index \(1\). We show that this spectral triple has simple dimension spectrum: For every operator \(B\) in the algebra \(\Psi(A,H,D)\) generated by the Shubin type pseudodifferential operators and the elements of \(A\), the zeta function \({\zeta}_B(z) = {\rm Tr} (B|D|^{-2z})\) has a meromorphic extension to \(\mathbb C\) with at most simple poles. Our main result then is an explicit algebraic expression for the Connes-Moscovici cyclic cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.

KW - Local index formulae

KW - Metaplectic operators

KW - Noncommutative orbifolds

KW - Spectral triple

UR - http://www.scopus.com/inward/record.url?scp=85136167516&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2008.11075

DO - 10.48550/arXiv.2008.11075

M3 - Article

VL - 409

JO - Advances in mathematics

JF - Advances in mathematics

SN - 0001-8708

IS - A

M1 - 108624

ER -