On Counting Twists of a Character Appearing in its Associated Weil Representation

Proceedings Mathematical Sciences

K Vishnu Namboothiri¹ 

[1] Department of Mathematics and Statistics, University of Hyderabad, Hyderabad 00 0, India$$

关键词: Nonarchimedian local field;    irreducible;    admissible representation of $GL(2;    F);    in$-factor of a character;    Weil representation;    number of characters appearing in its restriction.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Consider an irreducible, admissible representation 𝜋 of $GL(2,F)$ whose restriction to $GL(2,F)^+$ breaks up as a sum of two irreducible representations $𝜋_+ +𝜋_-$. If $𝜋=r_θ$, the Weil representation of $GL(2,F)$ attached to a character θ of $K^∗$ does not factor through the norm map from 𝐾 to 𝐹, then $𝜒in widehat{K^∗}$ with $(𝜒cdot p^{θ^{-1}})|F^∗=𝜔 K/F$ occurs in $r_{θ+}$ if and only if $in(θ𝜒^{-1},psi_0)=in(overline{θ}𝜒^{-1},psi_0)=1$ and in $r_{θ−}$ if and only if both the epsilon factors are $-1$. But given a conductor 𝑛, can we say precisely how many such 𝜒 will appear in 𝜋? We calculate the number of such characters at each given conductor 𝑛 in this work.

【 授权许可】

Unknown