All Questions
Tagged with buildings p-adic-groups
18 questions
1
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0
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75
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Cartan decomposition over a not-necessarily-discretely-valued field
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 ...
4
votes
0
answers
213
views
What are the good maximal compact subgroups in $p$-adic unitary groups?
Let $E/\mathbb Q_{p}$ be a quadratic extension and let $V$ be an $n$-dimensional $E$-hermitian space. Denote the hermitian form by $(\cdot,\cdot):V\times V \rightarrow E$. Let $G := \mathrm{U}(V)$ be ...
3
votes
1
answer
182
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Affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$
I saw the following results on affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$ without giving any references, where $\mathrm{SU}_3$ is the quasi-split inner form of special ...
3
votes
1
answer
261
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Is it possible to detect when a maximal parahoric subgroup is (hyper)special from its finite reductive quotient?
Let $F$ be a $p$-adic field with residue field $k$ and let $G$ be a connected reductive group over $F$. Let us assume that $G$ is simply connected as an algebraic group over an algebraic closure of $F$...
2
votes
0
answers
216
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Confusion regarding special parahoric subgroups of the unitary group
This question is to clarify some confusion about special parahoric subgroups of a unitary group $G = \mathrm U_n(F)$ in an odd number of variables, with respect to an unramified quadratic extension $E/...
5
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0
answers
122
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Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$
Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...
2
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0
answers
294
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Volume of double cosets $BwB$
In Macdonald's book "Spherical functions on a group of $p$-adic type", Prop. (3.1.7), it is stated that if $w=w_1\dots w_r$ is a reduced word for $w\in W$ (the affine Weyl group), and if $q(...
3
votes
0
answers
334
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Tits Reductive Groups over Local Fields Example 1.15 (Quasi-split special unitary groups in odd dimension)
I hope this question about Tits's paper "Reductive groups over local fields" in Algebraic groups and discontinuous subgroups ends up having an easy answer, but I'm a little stuck on the ...
3
votes
0
answers
61
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Directed galleries of the building of type $\widetilde{A}_{n}$
Let $X$ be the affine building of $GL_n(\mathbb{Q}_{p})$. We call oreinetd chamber of $X$ every sequence $\overrightarrow{C}=(s_1,...,s_n)$ of vertices such that $C=\{s_1,...,s_n\}$ is a chamber of $X$...
5
votes
0
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207
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Does $G$ act 2-transitively on its Bruhat-Tits building?
Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a semisimple Lie group over $k$. We consider the action of $G$ on its Bruhat-Tits building $X$.
Question: If $x,y,x',y'$ are vertices, ...
3
votes
1
answer
445
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When are maximal compacts same as maximal parahorics?
Let $G$ be a reductive algebraic group over a complete non-archimedean field $k$. We know that maximal compacts are exactly the same as maximal parahorics when the Iwahori is open compact subgroup of $...
9
votes
2
answers
671
views
What are the special parahoric subgroups in unitary groups?
Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special ...
5
votes
0
answers
574
views
How do you understand the Moy-Prasad filtration of G_2?
Starting on page 44 of this paper of Reeder and Yu, the authors describe the first graded piece of the Moy-Prasad filtration on $G_2$ at a certain point (in this case it's $GL_2$ of the residue field),...
4
votes
1
answer
579
views
Fixed points in the Bruhat-Tits building
Let $G$ be a connected reductive group over a complete discrete valuation field with perfect residue field (or just a non-arch local field). Let $\mathcal{B}$ be its reduced Bruhat-Tits building, and $...
5
votes
1
answer
638
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Arithmetic quotients of Bruhat-Tits buildings for groups over local fields of positive characteristic
I have been led to believe that there is a result giving a description of the quotient of a Bruhat-Tits building $\Delta(G,k)$, for a semisimple algebraic group $G$ over a non-archimedean local field ...
6
votes
1
answer
325
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When are toral orbits in buildings the difference of fixed-sets?
Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: "...
6
votes
1
answer
598
views
Clarification about Tits' article in the Corvallis
I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ (...
7
votes
1
answer
492
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When is a Moy-Prasad filtration subgroup the stabilizer of a subset of the building (up to center)?
Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an ...