Let G be a group and H a subgroup of G. Let $\pi_1$ and $\pi_2$ be irreducible representations of $G$ and $H$ respectively. By “branching laws” one refers to the rules of describing the vector space $Hom_{H} (\pi_1, \pi_2)$. The well known Langlands’ conjectures predict connections between the representation theory of reductive groups (over local and global fields) and the study of Galois representations. In the nineties, B. Gross and D. Prasad started a systemic investigations into the study of branching laws for the groups of interest to Langlands program, and their predictions are known as Gross-Prasad conjectures. We discuss two basic examples of these predictions. A covering group of a reductive groups is a certain central extensions. We discuss branching laws involving covering groups (namely, a two fold central extension of p-adic $GL_2$) which may be seen as a variation of Gross-Prasad conjectures for covering groups.

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Last updated: 24 Jun 2024