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Tagged Questions

A reductive group is an algebraic group $G$ over an algebraically closed field such that the unipotent radical of $G$ is trivial

3
votes
0answers
28 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 \...
2
votes
0answers
137 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, ...
2
votes
0answers
63 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
votes
2answers
274 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
votes
0answers
118 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 ...
2
votes
0answers
62 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 ...
3
votes
0answers
128 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 ...
1
vote
0answers
80 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
votes
1answer
144 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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vote
0answers
59 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 ...
1
vote
0answers
110 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
votes
1answer
106 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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vote
0answers
87 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)...
5
votes
1answer
191 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
votes
1answer
138 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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0answers
185 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
votes
1answer
144 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 ...
1
vote
0answers
37 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
votes
0answers
121 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 ...
3
votes
0answers
274 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
votes
1answer
203 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
votes
0answers
75 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
votes
1answer
146 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
votes
2answers
284 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
votes
1answer
264 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 $\...
0
votes
0answers
119 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
votes
1answer
187 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
votes
2answers
738 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
votes
0answers
217 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
vote
1answer
225 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
votes
1answer
109 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 ...
2
votes
1answer
42 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 $N^-$ ...
2
votes
0answers
74 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
vote
1answer
125 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
votes
0answers
65 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
145 views
+150

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
146 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 $\chi$ ...
7
votes
1answer
281 views

Representations of groups with the same derived group, how much control do we have over the central character?

Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
5
votes
0answers
96 views

Dimension of space of K-fixed vectors

If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular, (1) $H(G(...
8
votes
1answer
233 views

Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$

Let $G = \operatorname{GL}_n(F)$ for a $p$-adic field $F$, and let $G_D = \operatorname{SL}_n(F)$. I am wondering if there is a connection between irreducible, admissible representations of $G$ and ...
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votes
0answers
125 views

Are these statements correct for all reductive groups, or just for $\operatorname{GL}_2$?

I'm reading the following notes on Brian Conrad's website. There are a couple of statements there which seem too good to be true. There is Proposition 3.6, which states: (Bernstein): Every ...
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votes
0answers
80 views

Meaning of a highly ramified character for reductive groups

Let $F$ be a $p$-adic local field, and $G$ a connected reductive group over $F$. What is the meaning of a "highly ramified character" of $G(F)$? I have seen this terminology in many places in ...
1
vote
1answer
72 views

Reduced decomposition for Weyl group elements which support a Bessel function

Let $\Delta$ be a set of simple roots for a reduced root system, and let $(W,S)$ be the associated Coxeter system, where $W$ is the Weyl group and $S$ is the set of simple reflections corresponding to ...
5
votes
1answer
175 views

Generic supercuspidal representations of $\operatorname{GL}_n$ can be defined by integrals over $U$

Let $(V,\pi)$ be an irreducible, admissible, supercuspidal representation of $G = \operatorname{GL}_n(F)$ for $F$ a $p$-adic field. Let $B = TU$ be the usual Borel subgroup, maximal torus, and ...
11
votes
2answers
392 views

Split rank of inner forms

Let $G$ be a (connected) reductive group over some ground field $F$ and $G^*$ its unique quasi-split inner form. Denote by $\operatorname{rank}_F G$ the split rank of $G$, i.e. the dimension of a ...
6
votes
1answer
293 views

Rosenlicht's theorem and rationality questions

Let $G$ be a connected algebraic group over an algebraically closed field $\overline{k}$ acting on an irreducible variety $X$. A geometric quotient is a morphism of varieties $\pi: X \rightarrow X/\...
2
votes
1answer
152 views

Discrete decomposability of unitary representation

[INTRODUCTION] Let $G$ be a non-compact simple Lie group, and $G'$ a reductive subgroup of $G$. Suppose that $\pi$ is a non-trivial (hence, infinite dimensional) irreducible unitary representation of ...
3
votes
0answers
104 views

Holomorphic homomorphism of complex reductive groups is algebraic

Could you hint (or give a reference) for: A holomorphic group homomorphism between complex reductive algebraic groups is algebraic. Thank you!
3
votes
0answers
92 views

Kottwitz's vertical map

I'm looking at the map $w_H$ defined by Kottwitz in "Isocrystals with additional structure II" in section 7. It is a surjective group homomorphism defined for all reductive groups $H$ from $H$ to $X^{*...
0
votes
1answer
241 views

Existence of a model over S-integers

Let $G$ is a connected reductive group over a number field $F$. Let $S$ be a set of places containing the archimedean places for which $G$ has a model over the ring of $S$-integers in $F$. Is it true ...

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