Mathematical Research Letters
Volume 26 (2019)
Positivity of Shimura operators
Pages: 587 – 626
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