Questions tagged [reductive-groups]
A reductive group is an algebraic group $G$ over an algebraically closed field such that the unipotent radical of $G$ is trivial
211 questions
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19 views
Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is “uniform” across $P \overline{N}$
Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup ...
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64 views
Do we have $K \cap P = (K \cap M)(K \cap N)$?
Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...
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68 views
+100
Some confusion about weights and roots in parabolic root systems
I was reading James Arthur's book An Introduction to the Trace Formula and had a couple of questions. Here $A_0$ is a maximal split torus of a reductive group $G$, $P_0 \supset A_0$ is a minimal ...
4
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76 views
How to determine the unramified character corresponding to an unramified Langlands parameter?
Let $F$ be a p-adic field with ring of integers $\mathcal{O}$. Let $\textbf{G}$ be a connected split reductive algebraic group over $F$. For simplicity, we assume that $\textbf{G}$ is a Chevalley ...
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0answers
53 views
Volume of a double class of a parahoric subgroup
Let $F$ be a non-archimedean local field with residue field $F_q$. Let $G$ be the group of $F$-rational points of a connected reductive group defined and split over $F$. Fix a maximal split torus $T$ ...
2
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36 views
If $f \in \operatorname{c-Ind}_Q^P \sigma$, then $f|_N$ is compactly supported modulo $Q \cap N$?
There is a claim in a proof in Casselman's notes on representation theory which I have not been able to verify. I have asked several people and nobody knows why it is true.
The thing I can't figure ...
5
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1answer
205 views
$G(k)/H(k)$ as a submanifold of $G/H(k)$
Let $k$ be a local field (if necessary, assume characteristic zero). In general, if $X$ is a smooth variety of finite type over $k$ of dimension $n$, then the set of $k$-rational points $X(k)$ is an ...
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1answer
104 views
The quotients of double cosets $P_\theta \backslash P_\theta w P_\Omega$ are algebraic varieties over $k$
Let $k$ be a $p$-adic field, $G$ a connected reductive group over $k$ with minimal parabolic $P_0$ containing a maximal split torus $A_0$. Let $W = N_G(A_0)(k)/Z_G(A_0)(k)$ be the Weyl group, and $S \...
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1answer
81 views
If $\Pi$ and $\Sigma$ agree at almost all places, then the central character of $\Pi$ corresponds to $\operatorname{Det} \Sigma$
Let $\Sigma$ be an $n$-dimensional representation of the global Weil group $W_F$ for a number field $F$, and $\Pi$ an automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$. Suppose that at ...
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1answer
81 views
Reference for parabolic root systems
Let $G$ be a connected reductive group with maximal split torus $A_0$, and $P = MN$ a parabolic subgroup with Levi $M$ containing $A_0$. Let $A_M$ be the split component of $\mathfrak a_M^{\ast} = X(...
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30 views
Adjacent parabolic subgroups and proportionality to $\alpha^{\vee}$
Let $P = MN$ be a parabolic subgroup of a $p$-adic reductive group $G$ with split component $A_M$. There is bijection from the set of parabolic subgroups of $G$ with Levi $M$ and the chambers of $\...
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62 views
The convergence of the factor $\gamma(P)$ in the Iwasawa decomposition
Let $G$ be a connected, reductive group over a $p$-adic field $k$, $A_0$ a maximal split torus of $G$, and $P = MU$ a parabolic subgroup with $M$ containing $A_0$. Let $\overline{P} = M \overline{U}$ ...
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1answer
73 views
Integration over a reductive group $G$ using the constant $\gamma(P)$
Let $G$ be a connected, reductive group over a $p$-adic field. Let $A_0$ be a maximal split torus of $G$ and $P = MU$ a parabolic subgroup with Levi $M$ containing $A_0$, and opposite parabolic $\...
4
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2answers
150 views
Convergence of the intertwining operator as a vector valued integral
Let $G$ be a connected, reductive group over a $p$-adic field with parabolic $P = MN$ defined by a set of simple roots $\theta \subset \Delta$. For $(\pi,V)$ a representation of $M$, and $\nu \in \...
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152 views
Is a reductive group scheme always parahoric?
Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
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67 views
Classical reductive group schemes vs. unitary groups of separable algebras with involution — reference request
Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-...
7
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2answers
280 views
Explicit description of SU(2,2)/U
Consider the real diagonal $4\times 4$ - matrix
$$I_{2,2}={\rm diag}(1,1,-1,-1)$$
and the corresponding special unitary group
$$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I_{2,2}\...
4
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0answers
130 views
Families of Hessenberg varieties for $GL_n$
In short, the question is
What do we know about the sheaf $\pi_*\underline{\bar{\mathbb{Q}}_{\ell}}$ given by the family of (very original, see below) Hessenberg varieties for $GL_n$? As a sum of ...
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66 views
Valuations of root group elements appearing in the intersection of Iwasawa and Cartan double cosets
$\newcommand{\GL}{\operatorname{GL}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\val}{\mathit{val}}$Let $F$ be a local non-Archimedean field with valuation $\val$ and $G$ be (the
$F$-points ...
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0answers
195 views
Geometric interpretation of duality for representations of reductive groups
For a reductive group $G$ over a nonarchimedean local field, let $\Omega$ be a connected component of the variety of cuspidal data. Let $\Omega$ have dimension $d$. Let $\mathcal{M}^f(\Omega)$ be the ...
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93 views
Pairing half the sum of the roots with a simple coroot
I was calculating something with the root system $A_n$ and I think there might be a more general principle at work.
Here is the example: let $G = \operatorname{GL}_5$, with maximal torus $T$ and ...
4
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1answer
154 views
Do the absolute roots restricting to a given root form a Galois orbit?
Let $S$ be a maximal split torus of a connected, reductive group $G$. Let $P_0$ be a minimal $k$-parabolic containing $S$, $T$ a maximal torus of $P_0$ which is defined over $k$ and contains $S$, and ...
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0answers
68 views
Is the root cone is contained in the weight cone?
Originally posted on math.stackexchange. Let $G$ be a semisimple algebraic group over a field $k$. Let $A_0$ be a maximal split torus of $G$, and $\mathfrak a_0 = \operatorname{Hom}(X(A_0), \mathbb ...
2
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0answers
137 views
Errata for Casselman's unpublished notes
In the first chapter of W. Casselman's unpublished notes on representation theory, there is at least one stated result which is not true:
A counterexample to this last result is given in the question ...
2
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1answer
110 views
Minimal parabolic subgroups are $G(k)$-conjugate: a cohomological interpretation?
Let $G$ be a connected,reductive group over a $p$-adic field $k$. Let $M_0$ be a minimal Levi subgroup of $G$, and define $M_0^{\operatorname{der}}$ to be $M_0 \cap G_{\operatorname{der}}$. Lemma 2....
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97 views
Group schemes and Hyperspecial maximal compact subgroups
Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...
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1answer
199 views
Integral structures via lattices
I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ...
2
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1answer
141 views
semisimple support of character sheaves
So the essential question is:
How should we think about, or if possible compute, the semisimple
support of a cuspidal character sheaf?
For example, let $G=SL_2$. We have the cuspidal character ...
11
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0answers
187 views
Bernstein-Zelevinsky classification: viewing a representation as a subrepresentation or a quotient
$\DeclareMathOperator{\GL}{GL}$
$\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible ...
6
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1answer
150 views
Irreducibility of the unramified principal series
Let $G = \operatorname{GL}_n(F)$ with the usual Borel subgroup $P = TU$. Let $\chi = \chi_1 \otimes \cdots \otimes \chi_n$ be an unramified character of $T$. Suppose that $\chi$ is regular, which is ...
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0answers
39 views
A normalized embedding $\mathbb C \rightarrow \mathfrak a_M^{\ast}$ via $\tilde{\alpha}$
Let $G$ be a connected, reductive group over a field $k$. Let $S$ be a maximal $k$-split torus of $G$ with Weyl group $W$, $\Delta$ a set of simple roots of $S$ in $G$, and $P = MN$ a maximal ...
8
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0answers
134 views
Local Langlands Correspondence for unramified principal series representations
Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which ...
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0answers
283 views
On Local Langlands correspondences
Both over global function fields and $p$-adic fields, we have a series of conjectures under the name of “geometric Langlands conjectures”.
Over global function fields of char $p$, they are due to ...
3
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1answer
205 views
Possible groups appearing in a Shimura datum
Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for ...
5
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0answers
80 views
When is an irreducible unramified principal series representation $K$-spherical?
Let $G = \operatorname{GL}_n(\mathbb Q_p)$, $T$ the usual maximal torus of $G$, and $K = \operatorname{GL}_n(\mathbb Z_p)$.
Let $\chi$ be an unramified character of $T$, with $\chi(t_1, ... , t_n) =...
4
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1answer
155 views
Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper
In Kottwitz's 1985 Compositio paper,
Isocrystals with additional structure, first page, paragraph 4:
Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...
11
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2answers
295 views
How does the Bernstein-Zelevinsky construction of irreducibles from supercuspidals parallel the representations of the Weil-Deligne group?
In the Corvallis article Number Theoretic Background, here is what John Tate has to say on the local Langlands correspondence for a $p$-adic field $F$:
So, granting a correspondence between ...
8
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1answer
279 views
Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$
Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\...
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0answers
124 views
Local Langlands for $\textrm{GL}_n \times \textrm{GL}_m$
My question could apply more generally to a product of reductive groups over a $p$-adic field $k$.
Let $G_1 = \operatorname{GL}_{n_1}$ and $G_2 = \operatorname{GL}_{n_2}$. Any irreducible admissible ...
5
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1answer
189 views
Representations versus (g,K) modules
Let $G$ be a connected semisimple Lie group with finite center.
Let $(\pi,V)$ be an admissible representation on a Banach space $V$.
Is it true that the following are equivalent?
(a) $\pi$ is ...
14
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2answers
759 views
What is the archimedean Hecke algebra?
Let $\mathbf G$ be a connected, reductive group over $\mathbb Q$. For each nonarchimedean place $v$, let $K_v$ be a maximal compact subgroup of $\mathbf G(\mathbb Q_v)$. The space $\mathscr H(\mathbf ...
8
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0answers
219 views
Why the hyperoctahedral group is a ``reductive'' group?
Sorry for the misleading title, I actually mean the following:
The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...
1
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1answer
243 views
Complexification of compact Lie Groups and complex algebraic linear reductive groups
I'm studying complexifications of compact Lie groups on "Representation of compact Lie groups- Dieck Brocker".
I found on internet that there is a bijection between complexifications of compact Lie ...
2
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1answer
114 views
Contragredient of a cuspidal representation
Let $G$ be a reductive group over a nonarchimedean local field $F$. Let $\pi$ be an irreducible, cuspidal representation of $G$, with contragredient $\tilde{\pi}$. Then $\tilde{\pi}$ is cuspidal.
A ...
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1answer
46 views
Functions in the induced space compactly supported in $PN^-$ modulo $P$
Let $P_0$ be a minimal parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field $k$. Let $P$ be a parabolic subgroup containing $P_0$ with Levi decomposition $P = MN$. Let $N^-$ ...
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0answers
78 views
Is there a “big open cell” analogue for parabolic subgroups?
Let $G$ be a reductive group over a $p$-adic field. Let $P$ be a parabolic subgroup of $G$ containing a minimal parabolic $P_0$. Let $S$ be a maximal split torus of $P_0$, and let $\Delta$ be a set ...
1
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1answer
130 views
Is the Borel subgroup the only closed double coset?
Let $G$ be a quasisplit connected reductive group over a $p$-adic field $k$. Identify $G$ with its rational points. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$, both defined ...
2
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0answers
66 views
Continuity of the conductor of automorphic representations
I am interested in properties of (semi-)continuity of the conductor of automorphic representation. Let $F$ be a number field.
The meta question is, given a function on the unitary dual of $PGL(2, F)$ ...
4
votes
1answer
178 views
Making sense out of intertwining operators defined by a vector valued integral
Let $G$ be the rational points of a connected, reductive group over a $p$-adic field $F$. Let $S$ be a maximal split torus of $G$ with $\Delta$ a set of simple roots corresponding to a minimal ...
3
votes
1answer
151 views
For tori $S \subseteq T$, every character of $S(k)$ extends to a character of $T(k)$?
Let $k$ be a $p$-adic field, $T$ a torus over $k$, and $S$ an $k$-subtorus of $T$. If $\chi: S(k) \rightarrow \mathbb{C}^{\ast}$ is a smooth (resp. continuous) homomorphism, then does $\chi$ ...
