[Paper Review] Ramanujan's Master Theorem for Riemannian symmetric spaces

Gestur Ólafsson, Angela Pasquale|arXiv (Cornell University)|Mar 26, 2011

Advanced Algebra and Geometry20 references3 citations

TL;DR

This paper establishes a non-abelian generalization of Ramanujan's Master Theorem for Riemannian symmetric spaces of arbitrary rank, using the duality between compact and noncompact symmetric spaces within their complexification. The key result extends the classical Mellin transform interpolation formula to spherical Fourier transforms on symmetric spaces via holomorphic functions in a Hardy class, with explicit integral representations and duality-based inversion formulas.

ABSTRACT

Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of Riemannian symmetric spaces of compact and noncompact type inside a common complexification, we prove an analogue of Ramanujan's Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extend the results proven by Bertram for Riemannian symmetric spaces of rank-one.

Motivation & Objective

• To extend Ramanujan’s Master Theorem from the rank-one case to semisimple Riemannian symmetric spaces of arbitrary rank.
• To establish a duality-based analogue of the Mellin transform interpolation formula for spherical Fourier series on compact and noncompact symmetric spaces.
• To generalize the classical Ramanujan formula to reductive symmetric spaces using multivariable Hardy classes and Weyl group invariance.
• To provide a framework for integral interpolation of coefficients via holomorphic functions in a Hardy class on the complexified symmetric space.

Proposed method

• Leverages the duality between compact-type symmetric spaces $X_U = U/K$ and noncompact-type symmetric spaces $X_G = G/K$ inside their complexification $X_{\mathbb{C}} = G_{\mathbb{C}}/K_{\mathbb{C}}$.
• Uses spherical Fourier series on $X_U$ of the form $f(x) = \sum_{\mu \in \Lambda^+} (-1)^{|\mu|} d(\mu) a(\mu + \rho) \psi_\mu(x)$, where $a$ is holomorphic in a Hardy class $\mathcal{H}(A,P,\delta)$.
• Applies the spherical Fourier transform to relate the coefficients $a(\mu + \rho)$ to the transform of $f$ on $X_G$, using the duality between $X_U$ and $X_G$.
• Derives an integral representation via contour integration: $f(x) = \frac{1}{|W|} \int_{\sigma + i\mathfrak{a}^*} \left( \sum_{w \in W} a(w\lambda) b(w\lambda) \right) \varphi_\lambda(x) \frac{d\lambda}{c(\lambda)c(-\lambda)}$, valid for $\|H\| < P/\Omega$.
• Establishes the duality formula $\int_{X_G} f(x) \varphi_{-\lambda}(x) dx = \sum_{w \in W} a(w\lambda) b(w\lambda)$ for $\lambda \in T_\delta \cap T_{\Sigma,m}$, generalizing the classical Mellin inversion.
• Uses Laplace transforms on $\mathfrak{a}_{\mathbb{C}}^*$ to generate functions in the Hardy class $\mathcal{H}(A,P,\delta)$, ensuring appropriate decay and holomorphy.

Experimental results

Research questions

• RQ1 Can Ramanujan’s Master Theorem be generalized from rank-one to higher-rank semisimple Riemannian symmetric spaces using duality?
• RQ2 How can the spherical Fourier transform on symmetric spaces be used to interpolate coefficients of a spherical Fourier series via holomorphic functions?
• RQ3 What is the precise form of the integral representation and inversion formula for the spherical Fourier transform in the context of reductive symmetric spaces?
• RQ4 How do the functions $b(\lambda)$ and $c(\lambda)$, arising from Harish-Chandra’s theory, contribute to the duality formula?
• RQ5 What are the conditions under which a function on $X_U$ with coefficients from $\mathcal{H}(A,P,\delta)$ admits a holomorphic extension to $X_{\mathbb{C}}$?

Key findings

• The paper establishes a generalization of Ramanujan’s Master Theorem to semisimple Riemannian symmetric spaces of arbitrary rank, providing an integral representation of the spherical Fourier transform via contour integration.
• The key formula $f(x) = \frac{1}{|W|} \int_{\sigma + i\mathfrak{a}^*} \left( \sum_{w \in W} a(w\lambda) b(w\lambda) \right) \varphi_\lambda(x) \frac{d\lambda}{c(\lambda)c(-\lambda)}$ holds for $\|H\| < P/\Omega$, ensuring convergence and analyticity.
• The duality formula $\int_{X_G} f(x) \varphi_{-\lambda}(x) dx = \sum_{w \in W} a(w\lambda) b(w\lambda)$ is valid for $\lambda \in T_\delta \cap T_{\Sigma,m}$, extending the classical Mellin inversion to higher rank.
• The coefficients $a(\mu + \rho)$ are interpolated via holomorphic functions in the Hardy class $\mathcal{H}(A,P,\delta)$, with $A < \pi$, ensuring convergence and analytic continuation.
• The construction uses Laplace transforms on $\mathfrak{a}_{\mathbb{C}}^*$ to generate functions in $\mathcal{H}(A,P,\delta)$, providing a systematic method to produce admissible coefficient functions.
• The results are extended to reductive symmetric spaces by combining the semisimple case with a multivariable extension of the classical Ramanujan theorem, establishing Theorem 7.1.