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 $\textrm{ Gr}(p, \mathbb{ K}^n) = \textrm{ SU}(n,\mathbb{ K})/\text{ S}(\textrm{ U}(p,\mathbb{ K}) \times \textrm{ U}(n-p,\mathbb{ K}))$ 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) $\lambda =-1,\ldots, -p$. 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.
Categories: math.RT
