Newform theory for the full space via local Shimura correspondence and Waldspurger-type theorem

Shintani给出了与GL_n(F)的非分支主级数表示有关的球面Whittaker函数的显式公式，其中F是特征为零的非阿基米德局部场。Miyuchi推广了Shintani的工作，计算了与GL_n(F)的新形式(由Jaquet和Shalika定义的K_c固定)相关的Whittaker函数。这种新形式的空间是一维的，宫内用与K_c相关的Hecke代数刻画了BK_c(小于GL_n的子群)上的相关Whittaker函数。同样，我们考虑了GL_n(F)的Iwahori子群J上的Whittaker函数。模K_c的Hecke代数大于模J的Hecke代数。假设J-不动空间是一维的，我们试图刻画满GL_n(F)上的Whittaker函数。含有Iwahori不动向量的GL_n的不可约表示与Iwahori-Hecke代数的不可约有限维表示之间存在双射。我们的Whittaker函数与对应于Iwahori Hecke代数的一维表示的表示相关联。在与Ramla Abdellatif的另一个项目中，我们考虑了与SL_2(Q_P)的亚普2-覆盖相关的mod p个真Hecke代数。Laura Peskin描述了p=1mod(4)情形下的球面对应，并在这种情况下得到了Savin型局部Shimura对应。我们得到了所有奇素数的这样的对应关系。

Shintani gave an explicit formula for spherical Whittaker function associated to unramified principal series representation of GL_n(F) where F is a non-archimedean local field of characteristic zero. Miyauchi extends work of Shintani and computes Whittaker functions associated to newforms (fixed by certain K_c, defined by Jaquet and Shalika) for GL_n(F). The space of such newforms is 1-dimensional and Miyauchi uses Hecke algebra associated to K_c to give description of associated Whittaker function on BK_c (subgroup smaller than GL_n). In a similar spirit, we consider Whittaker functions associated to Iwahori subgroup J of GL_n(F). The Hecke algebra modulo K_c is bigger than the Hecke algebra modulo J. Assuming that J-fixed space is one-dimensional, we attempt to describe Whittaker function on full GL_n(F). There is a bijection between the irreducible representations of GL_n containing an Iwahori fixed vector and irreducible finite dimensional representations of the Iwahori Hecke algebra. Our Whittaker function are associated to representations that correspond to one-dimensional representations of the Iwahori Hecke algebra. This is joint with Moshe Baruch and Markos Karameris.In another project with Ramla Abdellatif, we consider mod p genuine Hecke algebras associated to metaplectic 2-cover of SL_2(Q_p). Laura Peskin describes the spherical ones in the cases p=1 mod(4) and obtains Savin-type local Shimura correspondence in this setting. We obtain such correspondence for all odd primes.