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Results tagged with automorphic-forms
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user 173538
An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
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Schur lemma and Whittaker functions
$\DeclareMathOperator\GL{GL}$Let $(\pi,V)$ be an infinite dimensional irreducible admissible representation of $\GL_2(\mathbb{Q}_p)$. Let us fix an element $v_0\in V$ and define a vector space
$$V_{v_ …
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Irreducibility criterion unramified principal series
Let $G$ be a reductive algebraic group defined over $\mathbb{Q}_p$ with maximal split torus $T_s$, Borel subgroup $B = TN$ and Weyl group $W(G,T_s)$. Let us consider the $\mathbb{Q}_p-$points of $G$ a …
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Supercuspidal, spherical and discrete series representation
Let $G$ be an algebraic group defined over $\mathbb{Q}$ with maximal unipotent radical $N$. Let $\pi$ be an admissible representation of $G(\mathbb{Q}_p)$, we say that this representation is supercusp …
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From Maass forms to automorphic forms of $SL_2(\mathbb{A})$
I'm learning basic stuff about automorphic forms, please, if anything I say is not true, excuse me.
Let $X = \Gamma\setminus \mathcal{H}$ be a modular curve and let $f(\tau)$ be a Maass cusp form. Sin …
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Powers of automorphic Eisenstein series
Let $G$ be a reductive group defined over $\mathbb{Q}$. Let $P$ be a standard parabolic subgroup of $G$ with Levi decomposition
$$P = MN.$$
We denote by $R_{disc,M}$ the discrete spectrum of $M$. Let …
- 103
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Constant coefficient of Eisenstein series
Let $\chi$ be a character of $\mathbb{Q}^{\times}\setminus\mathbb{A}^{\times}$. We define an induced representation of $Mp_2(\mathbb{A})\simeq SL_2(\mathbb{A})\times \mathbb{C}^1$,
$$I(s,\chi) := \{\P …
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