关于$p$ -adic Schrodinger表示上不变范数的刚性
On the rigidity of invariant norms on the $p$-adic Schrödinger representation
摘要：
受局部紧群$(\mathbb{Q}_p^d,+)$上的$\mathbb{C}_p$值傅里叶变换问题的启发，我们研究了Heisenberg群的$p$ -adic薛定谔表示上的不变范数.我们的主要结果是一个极小性和刚性性质的规范在一个家庭的不变规范参数化的格拉斯曼。这个族是在辛群作用下的超范数的轨道，通过交织算子作用。我们还证明了循环代数表示的普遍酉完备化的等价物的一般基本性质。结合刚性性质，我们能够证明薛定谔表示在该族中的任何范数中的完备化满足不可约性的强概念和舒尔引理的强版本。规范，可以形成为最大的一个有限数量的规范，从该家庭也进行了研究。最后，我们提出了一系列悬而未决的问题。
英文摘要：
Motivated by questions about $\mathbb{C}_p$-valued Fourier transform on the locally compact group $(\mathbb{Q}_p^d,+)$, we study invariant norms on the $p$-adic Schrödinger representation of the Heisenberg group. Our main result is a minimality and rigidity property for norms in a family of invariant norms parameterized by a Grassmannian. This family is the orbit of the sup norm under the action of the symplectic group, acting via intertwining operators. We also prove general fundamental properties of quotients of the universal unitary completion of cyclic algebraic representations. Combined with the rigidity property, we are able to show that the completion of the Schrödinger representation in any of the norms in that family satisfies a strong notion of irreducibility and a strong version of Schur's lemma. Norms that can be formed as the maximum of a finite number of norms from that family are also studied. We conclude this paper with a list of open questions.
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