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The classification of 1-dimensional representations, i.e. characters $G_K \to E^\times$, is a bit more complicated than you imply in your question. Any such character lands in $O_E^\times$ by compactn …

You are quite correct that the Langlands transfer map does not preserve cuspidality in general. E.g. if you take a modular form of CM type, coming from a Groessencharacter $\psi$ of some imaginary qua …

So you've seen that there are essentially three types of (g, K)-modules: finite-dimensional ones; principal series; and discrete series. The finite-dimensional ones don't interest us, since they are n …

Let $G$ be a reductive algebraic group (over some alg. closed field $k$ of char 0), and $H$ a subgroup such that $(G, H)$ is spherical (i.e., the Borel $B$ of $G$ has an open orbit on $G/H$). Then $k[ …

This argument does not work because the Killing form is not generally positive definite, so the orthogonal of a subspace wrt the Killing form is not necessary a complement of the subspace.

This question is answered pretty definitively by the following recent paper: Gee, Toby; Taïbi, Olivier, Arthur’s multiplicity formula for $\mathrm{GSp}_4$ and restriction to $\mathrm{Sp}_4$, J …

Let $E / F$ be a quadratic extension of nonarchimedean local fields (characteristic 0 if it matters), and $\pi$ an irreducible infinite-dimensional smooth representation of $GL_2(E)$. Let $B$ be the u …

There's an explicit description of maximal compact subgroups of all unitary groups over local fields (not necessarily quasi-split) in section 3 of this paper: Gan, Hanke, and Yu, "On an exact mass fo …

My question is about a local computation in the paper of Gelbart and Jacquet, "A relation between automorphic representations of GL2 and GL3", from 1978. Let $\sigma$ be an irreducible smooth complex …

[Expanding my earlier comment into an answer]: Self-dual cuspidal automorphic representations of $GL(n)$ have been classified by Jim Arthur, as one of the main results of his monograph on automorphic …

Trivially not, for cardinality reasons. Any function from the set of prime powers to $\mathbf{Z}$ extends uniquely to a multiplicative function $\mathbf{Z} \to \mathbf{Z}$, and there are obviously unc …

Let $G$ be a split semisimple algebraic group scheme over $\mathbf{Z}$ (I'm mostly interested in the case $G = Sp_4$). Let $V$ be an irreducible representation of the generic fibre $G_{\mathbf{Q}}$, …

Let $G = SO_5(\mathbf{Q}_p)$ be the split special orthogonal group over $\mathbf{Q}_p$, and $H \subset G$ the group $SO_4(\mathbf{Q}_p)$, embedded in the usual way as the stabiliser of an anisotropic …

This is much easier than it looks. The point is that any reductive group $G$ is isogenous to the product of its radical, which is its centre $Z(G)$, and its commutator subgroup, which is a semisimple …

Jared Weinstein's PhD thesis (http://math.bu.edu/people/jsweinst/jswthesis.pdf) is an excellent reference for this kind of thing. See section 3.4 in particular, where he computes the space $S_k(\Gamma …