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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 …

In the book Local Newforms for GSp(4), Roberts and Schmidt have defined a theory of "new vectors" for smooth representations of $GSp_4$ over a nonarchimedean local field $F$ with trivial central chara …

I've stumbled across an answer to this old question of mine so I'm going to answer it myself. The following paper: Johnson-Leung, Jennifer; Roberts, Brooks, Twisting of paramodular vectors, Int. J. Nu …

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 …

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.

Roberts and Schmidt have developed a theory of new vectors for generic irreducible smooth representations of $\operatorname{PGSp}_4(F)$ for $F$ a nonarchimedean local field, using the "paramodular sub …

I'm adding an answer to this very old question of mine, since someone just contacted me about it. The problem is rather comprehensively solved in this 2019 preprint of Taeko Okazaki: Takeo Okazaki, L …

I am not entirely sure what Clozel is trying to do here. He doesn't make it terribly precise what he means, and as you yourself have realised, if you read the article literally it seems that one case …

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 …

Theorem: Let $G$ be a reductive algebraic group over a local field $F$, let $K$ be any maximal compact subgroup of $G(F)$, and let $Z = Z(G)$. Then $K \cap Z(F)$ is the unique maximal compact subgroup …

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 …

Yes, this can be done. In recent years, Clozel, Harris, Taylor and others have shown how to attach Galois representations to sufficiently nice automorphic representations of $GL_n$ (for arbitrary $n$) …

In general, all we can say from "general abstract nonsense" is that if $\sigma$ is a subrepresentation of $Ind_P^G(\pi)$, then $\pi$ is a quotient of $J_N(\sigma)$; but you don't immediately get any f …

Let $\pi$ be the irreducible generic unramified representation of $Sp(W) $ that is a subquotient of $Ind(\chi_1, \dots, \chi_n)$. I think the key here is to realise that this does not exist for all …