[Paper Review] Positivity of Shimura operators

Siddhartha Sahi, Genkai Zhang|arXiv (Cornell University)|Jun 16, 2016

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper resolves a long-standing question by G. Shimura on the positivity domain of Shimura operators on Hermitian symmetric spaces by linking their eigenvalues to Okounkov's polynomials. It provides an explicit formula for the eigenvalues, proving that the positivity condition holds precisely when a certain symmetric function involving trigonometric and rational terms is non-negative, thereby characterizing the sets 𝒜 and 𝒢 explicitly via analytic and algebraic geometry techniques.

ABSTRACT

In [16] G. Shimura introduced a family of invariant differential operators that play a key role in the study of nearly holomorphic automorphic forms, and he asked for a determination of their extquotedblleft domain of positivity extquotedblright. In this paper we relate the eigenvalues of Shimura operators to certain polynomials introduced by A. Okounkov, which leads to an explicit answer to Shimura's questions.

Motivation & Objective

• To determine the domain of positivity for Shimura operators, a family of invariant differential operators central to the theory of nearly holomorphic automorphic forms.
• To answer G. Shimura's open question about the sets 𝒜 and 𝒢 defined by non-negativity of eigenvalues of these operators.
• To establish a precise characterization of the spherical unitary parameters via eigenvalue positivity conditions.
• To connect the spectral theory of Shimura operators to symmetric polynomials introduced by A. Okounkov, enabling explicit computation of eigenvalues.

Proposed method

• The authors relate the eigenvalues of Shimura operators to a family of symmetric polynomials Pλ(x; τ, α), which are q→1 limits of Okounkov's interpolation polynomials.
• They use the Harish-Chandra homomorphism to express eigenvalues as rational functions in the parameter x ∈ 𝔞*, reducing the positivity problem to analyzing the sign of a symmetric function S(x − α).
• The analysis proceeds by case distinction on the relative size of x1 and x2, using explicit formulas for qℓ,0(x) and S(x − α) in terms of sine and rational functions.
• A key step involves proving that the sequence of eigenvalues cℓ(x) is decreasing and converges to c(x), which implies equivalence between the positivity of all cℓ(x) and the positivity of c(x).
• The paper uses continuity and asymptotic analysis, particularly in the case x1 = α + 1, to verify the positivity condition in boundary regions.
• The solution is completed by analyzing the zero set of S(x − α) = 0, which defines the boundary of the positivity domain, and proving that this set is symmetric and converges to the coordinate axes as m → ∞.

Experimental results

Research questions

• RQ1 What is the precise domain of positivity for the eigenvalues of Shimura operators on Hermitian symmetric spaces?
• RQ2 How do the eigenvalues of the generating operators L1j relate to the spherical unitary parameters in the context of representation theory?
• RQ3 Can the positivity condition for all λ ∈ Λ be reduced to a condition on the fundamental generators L1j?
• RQ4 What is the geometric and analytic structure of the set where all eigenvalues remain non-negative?
• RQ5 How does the positivity domain behave asymptotically as the parameter m increases?

Key findings

• The eigenvalue cλ,x of the modified Shimura operator L′λ is given explicitly by cλ(x) = (−1)^|λ| η(Lλ), where η is the Harish-Chandra homomorphism.
• The positivity domain 𝒜 is characterized as the set of x ∈ 𝔞* for which the symmetric function S(x − α) is non-negative.
• The set 𝒢, defined by non-negativity of eigenvalues for the generators L1j, coincides with 𝒜, showing that the fundamental generators suffice to determine the full positivity domain.
• The boundary of the positivity region is defined by the zero set of S(x − α) = 0, which is symmetric about x1 = x2 and asymptotically approaches the coordinate axes as m → ∞.
• The critical point cm where the curve S(x − α) = 0 crosses x1 = x2 satisfies π cot(πcm) = ∑_{i=1}^m 1/(cm + i), and cm → 0 as m → ∞.
• The paper proves that the eigenvalue sequence cℓ(x) is decreasing and converges to c(x), which implies that the positivity of all cℓ(x) is equivalent to c(x) ≥ 0.