Partial Cosine-Funk Transforms at Poles of the ${\text{Cos}}^{\lambda }$ Transform on Grassmann Manifolds

Abstract

The cosine-$\lambda$ transform, denoted ${\mathcal{C}}^{\lambda }$, is a family of integral transforms we can define on the sphere and on the Grassmann manifolds $\text{Gr}(p,{\mathbb{K}}^n)=\text{SU}(n,\mathbb{K})/\text{S}(\text{U}(p,\mathbb{K})\times \text{U}(n-p,\mathbb{K}))$ where $\mathbb{K}$ is $\mathbb{R}$, $\mathbb{C}$ or the skew field $\mathbb{H}$ of quaternions. The family ${\mathcal{C}}^{\lambda }$ extends meromorphically in $\lambda$ to the complex plane with poles at (among other values) $\lambda =-1,\dots ,-p$. In this paper we normalize ${\mathcal{C}}^{\lambda }$ and evaluate at those poles. The result is a series of integral transforms on the Grassmannians that we can view as partial cosine-Funk transforms. The transform that arises at $\lambda =-p$ is the natural analog of the Funk transform in this setting.