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I'm trying to understand the last inclusion in the proof of Lemma 1.6.1 in Haines, Kottwitz, and Prasad's Iwahori-Hecke Algebras, which I will restate for convenience. We have $G$ as a split connected ...
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Let $G$ be a reductive p-adic group. $P$ is a parabolic subgroup of $G$ with Levi decomposition $P=MN$. Use $\bar{N}$ to denote the opposite of the unipotent radical $N$. The question I want to ask is ...
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Let $G$ be a reductive group over a $p$ adic field $F$ ($GL_2$ is good enough for me). Then there is the Bernstein classification of irreducible representations by cuspidal data and also the ...
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Let $G$ be an unramified reductive group over a non-archimedean local field $F$. Let $K\subset G(F)$ be a hyperspecial subgroup. It is known that if an irreducible representation $\pi$ of $G(F)$ over $...
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Let $G$ be a p-adic group, and use $\mathcal{H}(G)$ to denote the space of locally constant compactly supported functions on $G$. $\mathcal{H}(G)$ becomes an associative algebra under convolution. We ...
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Let $K$ be a local non-Archimedean field and $G$ a closed nondiscrete subgroup of $\mathrm{GL}_n(K)$, or a $K$-Lie analytic group (though I am primarily interested in linear groups). Is the following ...
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Let $G$ be a p-adic group, $H$ is a closed subgroup of $G$. Assume both are unimodular. Let $(\sigma, W)$ be a smooth representation of $H$, with its contragredient $(\sigma^\vee,W^\vee)$, with ...
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Let $p$ be prime and let $\mathbb{Z}_p$ be the ring of $p$-adic integers. Let $G$ be a (topologically) finitely presented pro-$p$ group, and let $F$ be a closed normal subgroup of $G$. Suppose that $F$...
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$\DeclareMathOperator{\Hom}{Hom}$Let $G$ be a p-adic group. Schur's lemma for smooth representations of $G$ states that if $\pi$ is an irreducible smooth representation of $G$, then $\dim \Hom_G(\pi,\...
user550518's user avatar
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Let $G=GL_n(F)$ over a p-adic field $F$, use $\mathcal{S}(G)$ to denote the compactly supported locally constant functions on $G$. Let $\Omega_n:=N_nA_n\omega_n N_n$ be the open dense Bruhat cell of $...
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I am trying to learn something about torsion-complete abelian $p$-groups. My reference is Fuchs, Infinite Abelian Groups, Vol II (Academic Press 1973). On page 15 he gives the setup. $B_n$ is a direct ...
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Let $\mathbb{Z}_p$ denote the $p$-adic integers. Does $\operatorname{GL}_2(\mathbb Z_p)$ have an open solvable subgroup $H $? By solvable I mean that there exist a chain of subgroups $$1=G_0\lhd G_1 \...
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I am studying the boundary of biregular trees, particularly those arising as Bruhat-Tits buildings, and I have a question about their explicit description, on par with what we know about regular trees....
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Fix a prime $p$ and let $G$ be a connected reductive group over a $p$-adic local field $K$. Let $H \subset G$ be a connected reductive subgroup such that $G(K)/H(K)$ is compact in the $p$-adic ...
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$\DeclareMathOperator\SL{SL}\newcommand{\Z}{\mathbb{Z}}\newcommand{\bbF}{\mathbb{F}}$I'm trying to understand F. Calegari's paper here, and there is a technical point I find confusing. Here's the ...
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Suppose I have a matrix over the $l$-adic integers $\mathbb{Z}_l$ which is diagonalizable over $\mathbb{Q}_l$. How to classify such matrices by similarity over $\mathbb{Z}_l$?
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Both in the global geometric Langlands as well as the local ($l \neq p$) Langlands formulated in the paper Geometrization of the local Langlands correspondence by Fargues and Scholze, there is a ...
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$\DeclareMathOperator\GL{GL}$Let $F$ be a local field and $\GL_n$ and $\GL_m$ defined over $F$. Let $\pi$ and $\sigma$ be an irreducible smooth representation of $\GL_n$ and $\GL_m$, respectively. Let ...
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In P.Cartier's article "Representations of $\mathfrak p$-adic groups: a survey" something is written at the very beginning of IV that I don't quite understand. Let $G$ be a $p$-adic ...
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Let $G$ be the group $GL(n,F)$, where $F$ is a p-adic field, and $I$ its standard Iwahori subgroup. Let $\pi$ be an irreducible smooth representation of $G$ with nonzero $I$-fixed vectors. It is well ...
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I'm trying to better understand the set $E$ of regular elliptic elements of $D^\times$, where $D$ is a finite dimensional central division algebra over a non-archimedean local field $F$. For example, ...
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Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity $w$ if for every homomorphism $ f:...
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For some calculations related to the unramified principal series of ${\rm GL}(n)$ over a $p$-adic field, I need to compute Hall-Littlewood polynomials that are associated to $n$-tuples that are not ...
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In his Cuspidal geometry of p-adic groups [J. Anal. Math. 47, 1–36 (1986)], Kazhdan uses a standard (?) result about representations of $p$-adic groups, which I will try to restate here. Let $F$ be a $...
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This is a follow-up to Does the bruhat decomposition induces decomposition on integral points (on an open cell)? Given a split connected reductive group $G$ over a $p$-adic local field $F$ with ring ...
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Let $G$ be a connected reductive group over a $p$-adic field $F$. For simplicity, we assume $G$ to be split. Fix a Borel $B=TU$ with its Levi decomposition ($U$ unipotent radical, $T$ maximal torus). ...
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Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and ...
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Let $G$ be $p$-adic group and let $G \rightarrow GL(V)$ be a representation. For example, $V$ is a quadratic $\mathbb{Q}_p$-space and $G$ is the associated orthogonal group. Take a point $v \in V$, ...
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In dimension 1 it can happen, that two p-adic groups share the same Bruhat-Tits building as the latter are highly symmetric trees. Can the same happen in higher dimensions as well? If it happens, is ...
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$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Cl{Cl}$[Reference: S. Lang, Cyclotomic Fields I and II, §2 chap 6] Let $K$ be a number field. Suppose that $K$ contains the $p$-th roots of unity if $...
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Let $G$ be a connected reductive group over a $p$-adic field $F$. Let $T\subset G$ be a maximal torus. Fix a special maximal compact subgroup $K$ of $G(F)$ and for any closed subgroup $H\subset G(F)$ ...
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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\det{det}\DeclareMathOperator\Ind{Ind}$Let $F$ be a $p$-adic local field and $\chi_i$ be unramified characters of $F^{\times}$. For a integer $m$, ...
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$\DeclareMathOperator\GL{GL}$For parabolic subgroup of a general linear group or classical group $G$, we can compute its modulus character using the positive roots associated to them. But it seems ...
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$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B} $My question is about the geometric lemma of $p$-adic classical groups not $\GL_n$. For $...
Andrew's user avatar
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Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix $$\begin{pmatrix} a & \varpi b \\ c & d \...
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Let $F$ be a local field of characteristic zero and $G$ a connected reductive group over $F$. Let us call an inducing datum a triple $(P,M,\sigma)$, where $P$ is a parabolic subgroup of $G$, $M$ is a ...
user449595's user avatar
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Representation theory and geometry over $k((t))$ and $\mathbb{Q}_p$ have many similarities, and there are many similar constructions, usually motivated from the other side (say the study of affine ...
Estwald's user avatar
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Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $\text{Jac}$ be the ...
Andrew's user avatar
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8 votes
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$\newcommand{\cts}{\mathrm{cts}}$Thanks for your reading. Let $A,B$ be two $\mathbb{Z}_p$-modules, where $\mathbb{Z}_p$ is the $p$-adic integer ring. I have two questions. Is $\mathrm{Hom}_{\mathbb{Z}...
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Let $\pi$ be an irreducible smooth representation of a classical $p$-adic group. Suppose that $\pi$ has a local L-parameter associated to some non-generic local A-parameter $\psi$. Then I am wondering ...
Andrew's user avatar
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Let $\mathcal{O}_{L}$ be the ring of integers of a finite extension $L$ of $p$-adic number fields $\mathbb{Q}_{p}$ where $p$ is an odd prime. Let $\mathfrak{m}_{L}$ be the maximal ideal of $\mathcal{O}...
stupid boy's user avatar
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Let $F$ be a $p$-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ ...
Andrew's user avatar
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$\DeclareMathOperator\Mat{Mat}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers with a uniformizer $\pi$. Let $|\cdot|\colon \mathbb{F}\to \mathbb{R}$ be ...
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If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
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$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...
asv's user avatar
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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. The natural representation of the group $\GL_n(\...
asv's user avatar
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Let $F$ be a $p$-adic field and $G_1=\mathrm{SL}_2(F)\subset \mathrm{GL}_2(F)=G$. For simplicity, we assume $p>2$. Denote by $|\cdot|$ the normalized absolute value on $F$. Here I shall focus on ...
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Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that $\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
Jeremy Rouse's user avatar
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$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}$ Let $F$ be a $p$-adic field and $\Sp(2n)$ symplectic group over 2n dimensional symplectic space over $F$. Let $B=...
Andrew's user avatar
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$\DeclareMathOperator\GL{GL}$Let $F$ be a $p$-adic field and $\chi$ be an unramified character of $\GL_1(F)$. Consider an induced representation $\pi$ of $\GL_2(F)$ induced from the character $\chi|\...
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