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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

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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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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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+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 ...
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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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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 $T$ ...
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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 ...
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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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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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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 $\...
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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 $K$-...
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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}\...
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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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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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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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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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 ...
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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 ...
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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) =...
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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 ...
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2answers
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 $PGL(2, F)$ ...
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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
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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 $\chi$ ...