Symplectic supercuspidal representations and related problems

Abstract

In this paper, we summarize the basic structures and properties of irreducible symplectic supercuspidal representations of GLn(F ) over a p-adic local field F with characteristic zero, and explore possible topics for further investigation.

Generalized Shalika model on $\mathrm{SO}_{4n}(F),$ symplectic linear model on $\mathrm{Sp}_{4n}(F)$ and theta correspondence

“…It turns out that these two Langlands quotients are related through theta correspondence. This fact fits nicely into interpretation of symplecticity of representations of GL 2n (F ) in terms of various functorialities and models existing on members of a dual pair (O 4n (F ), Sp 4n (F )); this is nicely explained in [8], p. 541.…”

“…In this note, we answer a question of Jiang posed in [8], p. 542. Namely, as we mentioned above, in the specific cases of induction from an irreducible supercuspidal symplectic representation of GL 2n (F ), the corresponding Langlands quotients, which have a non-zero generalized Shalika model, and a non-zero symplectic linear model, respectively, are related through the theta correspondence.…”

“…This result suggests that the functorialities mentioned in the preceding paragraph (cf. [8], p. 541) can be generalized in an appropriate setting, raising further questions about Gelfand-Graev models and Fourier-Jacobi models of the representations of SO 4n (F ) and SO 4n (F ) we have studied.…”

Women in Numbers Europe

“…For the general definitions we refer to Jiang et al (2010a) Section 2. Let F=Q p be a finite extension and fix a non-trivial additive character W F !…”

Model transition under local theta correspondence

“…In [JNQ10a], Jiang, Nien and Qin proved that under the local theta correspondence, for certain representations, the generalized Shalika model on SO 4n (F ) is corresponding to the symplectic linear model on Sp 4n (F ), and conjectured that it is true for general representations (see [JNQ10b,p. 542]).…”