Mathematical Research Letters

Volume 26 (2019)

Number 2

Positivity of Shimura operators

Pages: 587 – 626



Siddhartha Sahi (Department of Mathematics, Rutgers University, New Brunswick, New Jersey, U.S.A.)

Genkai Zhang (Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, Göteborg University, Göteborg, Sweden)


In this paper we answer some questions of G. Shimura on the positivity of certain invariant differential operators that play a key role in the study of nearly holomorphic automorphic forms in higher rank. Our main result expresses the eigenvalues of Shimura operators in terms of certain polynomials introduced by A. Okounkov, which leads to an explicit answer to Shimura’s questions.

Research by Genkai Zhang was partially supported by the Swedish Science Council (VR). Siddhartha Sahi was partially supported by an AIP research grant from Rutgers University and by the Simons Foundation.

Received 19 September 2017

Accepted 15 March 2018

Published 12 August 2019