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Are you asking for a proof of existence, or an explicit construction? These are very different things! It is immediate from the definition that there exists a finite family $(W_i, W_i')_{i \in I}$ wit …

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 …

No, there is not a well-defined subgroup "$G(\mathcal{O}_K)$" for a semisimple algebraic group over $K$; if you define it using embeddings into $GL_n$ then the subgroup you get will depend on the embe …

Let $G$ be a split reductive group over a nonarchimedean local field $F$ (I'm particularly interested in the case of $\operatorname{GSp}_{2n}$). Given a parahoric subgroup $K \subset G(F)$, and a pa …

I'm afraid that this question has a disappointingly simple answer. Yes, the values of the Iwahori-spherical Whittaker functions have an interpretation as characters of representations; but they are on …

To understand this question better one should remember what Langlands parameters actually are. A Langlands parameter isn't just a list of numbers: these numbers are the components of a map from some a …

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 …

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 …

This condition comes up because of $(\mathfrak{g}, K)$-cohomology, which is an extremely important invariant of automorphic representations. If $\xi$ is an algebraic rep, then $\xi$ defines a locally- …

Don't confuse $(\phi, N)$-modules (which are finite-dimensional vector spaces over $\mathbf{Q}_p$ with various extra structures) with $(\phi, \Gamma)$-modules (which are modules over a much bigger and …

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 $E / F$ 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 …

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 …

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 …

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[ …