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Does an induced representation commute with complexification?

Let $G$ be a compact Lie group, $K$ its closed subgroup, $\rho$ a finite-dimensional real irreducible representation of $K$, $c\rho$ its complexification, $\mathrm{Ind}^G_K(\rho)$ the real ...
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Is every irreducible representation of $Sp(m)$ a representation of $U(2m)$?

If $Sp(m)$ is the group of linear automorphisms of $\mathbb{C}^{2m}$ which fix a complex symplectic form $\omega$ and a quaternionic structure $j$ on $\mathbb{C}^{2m}$, then it is clear that $Sp(m)$ ...
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Boundedness of dimension of representations that restrict to a fixed representation of a normal subgroup

Let $G$ be a compact Lie group, $H$ a closed subgroup and $W$ an irreducible real representation of $G$. Then it follows from Frobenius reciprocity and Bott’s definition of induced representation that ...
rick's user avatar
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Minimal subrepresentation of the Weyl group

Background I want to generalize Theorem 3.2.1 in Dat, J., Orlik, S., & Rapoport, M. (2010). Period Domains over Finite and p-adic Fields (Cambridge Tracts in Mathematics). Cambridge: Cambridge ...
EJB's user avatar
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Geometric induction of modules for algebraic groups

Let $\Bbbk$ be an algebraically closed field (of any characteristic). Let $G$ be an algebraic group over $\Bbbk$, and $H$ a closed (hence algebraic) subgroup of $G$. Let $V$ be a finite-dimensional $...
freeRmodule's user avatar
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1 answer
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Intertwining operators and induced representation

Let $G=GL(n, F)$, $B$ be a Borel subgroup and let $B=AN$ be the Langlands decomposition. Let $\nu \in \mathfrak{a}^*_{\mathbb{C}}$ be in the positive Weyl chamber. Consider the normalized induced ...
random123's user avatar
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Explicit cochain for Shapiro's lemma with trivial coefficients

(cross-post from stack exchange after not receiving any answers) I'm wondering the following: if we have a finite-index subgroup $H\subset G$, and a cocycle $[c]\in H^1(H,\mathbb{Z})$, is there any ...
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How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?

It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
IntegrableSystemsEnthusiast's user avatar
2 votes
1 answer
148 views

How do you construct elements in $\operatorname{Ind}_P^G\pi$?

Let $G$ be a $p$-adic reductive group, and $P=MN\subseteq G$ a parabolic subgroup. How do you know that the space of the induced representation $\operatorname{Ind}_P^G\pi$ is non-zero? Namey, how do ...
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Evaluations of group characters on cosets of subgroups

Let $G$ be a finite group, $H$ a subgroup of $G$ and $g \in G$. Define $$ [gH] = \sum_{h \in H} gh, $$ viewed an element in the group algebra $\mathbb{C}[G]$. Given an irreducible character $\chi$ of $...
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Zeroes of characters of general linear group induced from certain characters of parabolic subgroups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}$My question is about the types of conjugacy classes of $\GL(n,q)$, the general linear group over the finite field with $q$ elements, on which ...
mathseeker's user avatar
5 votes
1 answer
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Tensoring with an induced representation: proof question

Let $G$ be a locally compact Hausdorff group and $H$ a closed subgroup of $G$. If $\sigma: H \to B(\mathcal{K}_\sigma)$ is a unitary representation of $G$, we can associate an "induced ...
Andromeda's user avatar
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Geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group

What is the geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group? Could Springer theory of Weyl group representations be used to obtain such a geometric meaning?
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Does the following corollary of Mackey's tensor product theorem hold for smooth representations?

Let $G$ be a locally profinite group, and let $H$ be a closed subgroup of $G$. Let $\sigma$ be a smooth representation of $G$, and let $\tau$ be a smooth representation of $H$ (henceforth, every ...
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Induction for quantum group

I am confused about a claim in the article Representation of quantum algebras by Henning Haahr Andersen, Patrick Polo and Wen Kexin. I probably misunderstood a definition, but I found two claims about ...
Nicolas Hemelsoet's user avatar
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The norm of the principal series intertwining operator for $\operatorname{GL}_2$

Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}_2(\mathbb Q_p)$? Background: For a character $\chi = (\chi_1,\chi_2)$ of the ...
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Irreducible Representation of A_5

Knowing the fact that standard representation arising out of permutation representation of $A_5$ over $\mathbb{C}$ is irreducible and of degree $4$. What can we conclude about the irreducibility over ...
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Jacquet module of unramified principal series representaion with respect to parabolic subgroup of $GL_n(F)$

Let $F$ be a local field of characteristic zero and $G=\operatorname{GL}_n(F)$. Let $B=UT$ be a Borel subgroup of $G$ and $\chi=(\chi_1,\cdots,\chi_n)$ is an unramified character of $B$. Consider ...
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Irreducibility of parabolic induction on unitary group

Let $F$ be a local field of characteristic 0. I know that $\pi=\text{Ind}_{B_k(F)}^{GL_k(F)}(\chi_1 \boxtimes \cdots \chi_k)$ for some unramified characters $\chi_i$'s is irrducible if there is no $\...
Monty's user avatar
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4 votes
2 answers
419 views

A morphism intertwining two induced representations

TL;DR: Given representations $D,\Lambda$ of subgroups $K,Q$ of a Lie group $G$, is it true that every intertwining operator $T$ between the resulting induced representations of $G$ can be written $$ (...
Michael_1812's user avatar
4 votes
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Mackey theory application - semidirect product abelian-by-finite

In order to advance my research I'm supposed to understand this fact: Let $A$ be an abelian group and $S$ a finite group acting on $A$. This defines the semi-direct product $A\rtimes S$: Let $\chi$ ...
Arnon Hod's user avatar
6 votes
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161 views

Generalisation of the Witt–Berman induction theorem

$\DeclareMathOperator{\Aut}{Aut}\DeclareMathOperator{\Ind}{Ind}$I believe I can prove the following induction theorem (modulo carefully checking a few details), and I would like to know whether this ...
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Embedding of discrete series

Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals that of its maximal compact subgroup. Let $G'$ be a reductive subgroup of $G$ with equal rank. If $\tau$ is a discrete ...
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Induced $(\mathfrak{g},K)$-modules

Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
Hebe's user avatar
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1 answer
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Two basic questions on parabolic induction

I want to ask some basic two questions on the parabolic induction. Let $F$ be a local fields. Let $\chi_1,\chi_2$ be two characters of $GL_1(F)$ and $GL_1 \times GL_1$ be the Levi part of the ...
Monty's user avatar
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Some basic question on the parabolic induction

I would like to ask some basic question about parabolic induction. Let $F$ be a local field and $G=GL_n(F)$ and $P=MN$ its parabolic subgroup whose Levis subgroup $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...
Monty's user avatar
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Index of subgroup generated by characters induced from $p$-elementary subgroups in the ring of virtual characters

I posted this over on MSE, but received absolutely no love. So maybe I’ll have better luck here. It seems like a relatively easy group theory question that I’m just not seeing! It’s on the essential ...
Nicholas Camacho's user avatar
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1 answer
859 views

Reference request: tensor induction

While working on a problem, I constructed something which looked like an induced representation, but with a tensor product instead of a direct sum. Here is a special case. Let $G$ be a group, with ...
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In what way are Joseph`s completion functors analogous to algebraic principal series modules of Dixmier?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $x \in W$ be an element of the Weyl group (which acts on weights by $x \cdot \lambda = x(\lambda + \rho) - \rho$, $\rho$ being the half sum of ...
Henrique Tyrrell's user avatar
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2 answers
741 views

Reference Request: Definition of Induced Representation for reductive groups over a local field

Let $G$ be a connected, reductive group over a local field $F$ of characteristic zero, and $H$ a closed subgroup of $G$ which is defined over $F$. Let $\mu_H, \mu_G$ be right Haar measures on $H(F), ...
D_S's user avatar
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9 votes
2 answers
725 views

Characters of irreducible unitary representations of the Poincaré group

Consider Poincare group $\mathrm{ISO}(1,d-1)$, given by $\mathbb R^{1,d-1}\rtimes SO(1,d-1)$ in signature $(1,d-1)$, for some odd $d \geq 3$. Denote the universal cover of the component connected to ...
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1 answer
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Irreducible representations occuring in $\mathrm{Ind}_G^{S_{|G|}}1$ for $G$ finite group

Let $G$ be a finite group with $|G|=n$, let $S_G=S_n$ be the group of $n!$ permutations of the set $G$. Then $G$ is a subgroup of $S_G$ via left-translation (i.e. $g\in G$ corresponds to the ...
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1 answer
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Does the induced representation preserve norm?

Let $G$ be a finite group, $N$ its normal subgroup, and $\pi:N\to{\mathcal B}(X)$ a unitary representation of $N$ in a Hilbert space $X$. Consider the induced representation $\pi':G\to{\mathcal B}(L_2(...
Sergei Akbarov's user avatar
2 votes
0 answers
354 views

A generalization of the notion of induced representation

Let $G$ be a Lie group which is a finite extension of an open normal subgroup $N$: $$ 1\to N\to G\to F\to 1 $$ (so $N$ and $G$ are Lie, and $F$ is finite; but I think, this is not very important, we ...
Sergei Akbarov's user avatar
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0 answers
256 views

$L^2$ norms of Whittaker vectors and zeros of Intertwining operators

For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard ...
Subhajit Jana's user avatar
3 votes
1 answer
358 views

Induced representation of locally compact groups

I am not having the best luck getting this, so any advice is appreciated. I apologize in advance if the question is too low-level. Let $G$ be a locally compact group $\sigma$-compact group, and $H\leq ...
Math-user's user avatar
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8 votes
1 answer
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Irreducible $S_n$-modules and $S_n$-actions on projective spaces

Let $V$ be an $(N+1)$-dimensional vector space with an action of the symmetric group $S_n$, such that $V$ is an irreducible $S_n$ -module. Let $\{p_1,...,p_h\}\in \mathbb{P}(V)$ be $h\geq N+2$ ...
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0 answers
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The relation of the local principal representations of $U(2)$ and $GL(2)$

Let $E/F$ be a quadratic extension of number fields and $v$ is a non-archimedean place of $F$. Let $G=U(2)(F_v)$ be the $F_v$-points of the 2-dimension unitary group associated to $E_v/F_v$ and $B$, $...
Monty's user avatar
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4 votes
1 answer
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convergent series representation for p-adic complex numbers

The field $\mathbb{C}_p$ of $p$-adic complex numbers is the completion of the algebraic closure of $\mathbb{Q}_p$ with the corresponding extension of the usual non-Archimedean valuation $|\;\;|_p$. ...
Chilote's user avatar
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5 votes
1 answer
467 views

Explicit formula of base change for GL(n)

Let $E/F$ be a quadratic extension of number fields and $v$ is a place of $F$. Let $\chi_1,\chi_2$ be the unramified characters of $F_v^{\times}$. If $B(\chi_1,\chi_2)$ is the unramified principal ...
Monty's user avatar
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3 votes
2 answers
565 views

Generic irreducibility of parabolic induction

In J.Bernstein's notes: REPRESENTATION OF P-ADIC GROUPS, he remarked the following result(see P.88): Let $G$ be a reductive group defined over nonarchimedean local field $F$, $P$ parabolic subgroup of ...
chluo's user avatar
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6 votes
1 answer
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Relation between Different Definitions of Induced Representation

I've seen two different ways to define induced representation. One is as in the book Introduction to representation theory: If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of $...
Megan's user avatar
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2 votes
2 answers
163 views

When is the induced representation factored through the initial one?

Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ ...
Sergei Akbarov's user avatar
4 votes
1 answer
1k views

When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?

This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected. It seems that the ...
Manuel Bärenz's user avatar
3 votes
1 answer
207 views

Inclusion of copies of an irrep as orthogonal subspaces of an induced representation

Suppose we have a finite group $G$ with subgroup $H$, a representation $\rho_V$ of $H$ on a finite-dimensional vector space $V$, and an $H$-invariant inner product on $V$: $$\forall x,y\in V, h\in H,\...
Greg Egan's user avatar
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1 vote
1 answer
123 views

Is it necessary for $\pi:H\to U(\mathcal{H}_{\pi})$ to be a homomorphism in order for $\text{ind}_H^G\pi$ to be weakly continuous?

Suppose $G$ is a locally compact group and $H$ is an open subgroup for simplicity. Further suppose $\pi$ is a representation of $H$ on some Hilbert space $\mathcal{H}_{\pi}$, i.e. $\pi(h)$ is unitary ...
Cameron Williams's user avatar
3 votes
0 answers
412 views

Decomposition of a representation of SU(N) into representations of SU(N-1)

Let $\omega_k$ be the highest weight of the $k$-th antisymmetric representation of $\mathfrak{su}(N)$. Consider an irreducible representation of $\mathfrak{su}(N)$, characterized by its highest ...
Bruno Le Floch's user avatar
3 votes
1 answer
299 views

‘Non-Induced’ Left Regular Representations of $ C^{*} $-Dynamical Systems

In what follows, a ‘$ * $-representation’ always means a non-degenerate $ * $-representation. Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and let $ \pi: \mathscr{A} \to B(\mathcal{...
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19 votes
3 answers
3k views

Finite groups such that every irrep can be induced from trivial irrep of a subgroup ?

What are examples (general features) of the finite groups $G$, such that every irrep (irreducible representation) is contained (as constituent) in the representation induced from trivial ...
Alexander Chervov's user avatar
10 votes
6 answers
1k views

When k[G/H] is multiplicity free G module ?

Consider finite group G and its subgroup H, and representation of G in k[G/H] i.e. functions on G/H. Question: What is known about the question: when k[G/H] is multiplicity free ? (Let us consider k ...
Alexander Chervov's user avatar