格拉斯曼流形上的Radon变换、余弦变换和正弦变换
Radon, cosine and sine transforms on Grassmannian manifolds
摘要：
设$G_{n,r}(\bbK)$是$\bbK^n$中$k$维$\bbK$ -子空间的格拉斯曼流形，其中$\bbK=\mathbb R, \mathbb C, \mathbb H$是实数域、复数域或四元数域.我们考虑了从$L^2$空间$L^2(G_{n,r}(\bbK))$到空间$L^2(G_{n,r^\prime}(\bbK))$的Radon变换、余弦变换和正弦变换，$\mathcal R_{r^\prime, r}$、$\mathcal C_{r^\prime, r}$和$\mathcal S_{r^\prime, r}$，对于$r, r^\prime \le n-1$。$L^2$空间被分解成$G$的不可约表示，且重数是自由的。我们在分解下计算变换的谱符号。为此，我们证明了紧环面$\mathbb R^r/{2πQ^\vee}$上正弦和余弦型函数在BC型一般根系上的两个Bernstein-Sato型公式，推广了我们最近关于非紧空间$\mathbb R^r$上双曲正弦和余弦函数的结果.然后，我们还发现了变换的图像的特征。我们的结果推广了Alesker-Bernstein和Grinberg的结果。我们进一步证明了Knapp-Stein交织算子对于某些诱导表示是由正弦变换给出的，并给出了Stein互补级数在紧图中的酉结构.
英文摘要：
Let $G_{n,r}(\bbK)$ be the Grassmannian manifold of $k$-dimensional $\bbK$-subspaces in $\bbK^n$ where $\bbK=\mathbb R, \mathbb C, \mathbb H$ is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, $\mathcal R_{r^\prime, r}$, $\mathcal C_{r^\prime, r}$ and $\mathcal S_{r^\prime, r}$, from the $L^2$ space $L^2(G_{n,r}(\bbK))$ to the space $L^2(G_{n,r^\prime}(\bbK))$, for $r, r^\prime \le n-1$. The $L^2$ spaces are decomposed into irreducible representations of $G$ with multiplicity free. We compute the spectral symbols of the transforms under the decomposition. For that purpose we prove two Bernstein-Sato type formulas on general root systems of type BC for the sine and cosine type functions on the compact torus $\mathbb R^r/{2πQ^\vee}$ generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space $\mathbb R^r$. We find then also a characterization of the images of the transforms. Our results generalize those of Alesker-Bernstein and Grinberg. We prove further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Stein's complementary series in the compact picture.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
