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Let‘s take $G=GL_n$ over a number field $F$ for example. It's already known that every irreducible automorphic representation $\pi$ is a irreducible component of a induced representation $I(G,P;\...

Let $\chi$ be a character of $\mathbb{Q}^{\times}\setminus\mathbb{A}^{\times}$. We define an induced representation of $Mp_2(\mathbb{A})\simeq SL_2(\mathbb{A})\times \mathbb{C}^1$, $$I(s,\chi) := \{\...

In Shimura's paper "ON THE HOLOMORPHY OF CERTAIN DIRICHLET SERIES", he constructed a family of Eisenstein series $E(z,s)$ by summing factor of automorphy. $E(z,s)$ is of negative half-integral weight, ...