Hall 123

Restriction of representations and metaplectic groups

Shiv Prakash Patel

IIT Bombay

​Restricting a ​representation of a group to some of its ​"nice"​​ subgroups is a way to understand the representation, ​e​specially ​to those subgroups ​which ​give multiplicity one​ or finite multiplicity​. I will discuss a few cases ​of restriction ​​in the study of p-adic groups. ​One case will be restricting​ ​the​ representations of ​$GL_2(E)​$​ to ​$​GL_2(F)​$​ where $E/F$ is a quadratic extension of $p$-adic fields. ​In a work of D. Prasad, this ​restriction ​is related to​ another restriction namely,​ ​the ​rest​r​iction ​of representations of $GL_2(E)$ ​to ​$​D_​​F​^{\times}$​, where ​$D_F$​ is the quaternion division algebra over $​F​$​​ and $D_F^{\times} \hookrightarrow GL_2(E)$​. ​ ​Prasad​ relates ​these two restrictions by a "dichotomy" ​involving the​ Jacquet-Langlands correspondence. In another case, we ​will ​discuss how a similar restriction​s​ can be studied in the case of ​the ​metaplectic cover of GL_2(E)​, which is not a linear group. ​