论文标题
$ \ operatorname {gl} _2（\ mathbf {q} _p）$的Mod- $ p $表示理论
The mod-$p$ representation theory of the metaplectic cover of $\operatorname{GL}_2(\mathbf{Q}_p)$
论文作者
论文摘要
在所谓的Metaplectic覆盖物上，自然将半综合重量模块化形式视为自动形式，$ \ operatotorname {gl} _2（\ Mathbf {\ MathBf {a} _ {\ Mathbf {q}}}）对于一个奇数的$ p $，我们对$ \ operatorname {gl} _2 _2（\ Mathbf {q} _p）$的相应覆盖的平滑不可约的真实mod- $ p $表示完整分类。 Pro- $ P $ IWAHORI HECKE代数。作为我们研究普遍模块在球形Hecke代数上不可约的亚序列的应用，具体取决于一定的重量，我们证明，有限的长度等同于有限产生和可接受。最后，我们解释了与局部代数不可证实的未命名的真实主要序列表示形式的关系。
Half-integral weight modular forms are naturally viewed as automorphic forms on the so-called metaplectic covering of $\operatorname{GL}_2(\mathbf{A}_{\mathbf{Q}})$ -- a central extension by the roots of unity $μ_2$ in $\mathbf{Q}$. For an odd prime number $p$, we give a complete classification of the smooth irreducible genuine mod-$p$ representations of the corresponding covering of $\operatorname{GL}_2(\mathbf{Q}_p)$ by showing that the functor of taking pro-$p$-Iwahori-invariants and its left adjoint define a bijection onto the set of simple right modules of the pro-$p$ Iwahori Hecke algebra. As an application of our investigation of the irreducible subquotients of the universal module over the spherical Hecke algebra depending on some weight, we prove that being of finite length is equivalent to being finitely generated and admissible. Finally, we explain a relation to locally algebraic irreducible unramified genuine principal series representations.
