# Questions tagged [induced-representations]

The induced-representations tag has no usage guidance.

55
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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 ...

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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 ...

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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 $...

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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 ...

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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 ...

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### 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 ...

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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 ...

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187
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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 ...

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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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450
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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 $\...

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### 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
$$
(...

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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 $\chi$ ...

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### 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 ...

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538
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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 ...

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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 $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...

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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 ...

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### 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 ...

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### 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), ...

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### 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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### 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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### 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(...

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### 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 ...

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### $L^2$ 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 ...

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### 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 ...

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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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### 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 $B$, $...

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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 $|\;\;|_p$.
...

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### 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 ...

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### 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 ...

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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 $...

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### 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 $\rho:G\to B(Y)$ ...

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### 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 ...

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### 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,\...

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### 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 ...

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### 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 ...

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### ‘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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### 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 ...

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### 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 ...