Questions tagged [induced-representations]

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79 views

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 $\chi$ ...
5
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0answers
83 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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0answers
77 views

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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81 views

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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141 views

Two basic question 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 ...
3
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116 views

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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0answers
221 views

Some question on the induced representation of tensor product

I would like to some question concering the induced representation of tensor product. Let $F$ be a local fields of charateristic 0 and $0<a<m$ are two positive integers. For $1 \le i \le m$, ...
3
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0answers
42 views

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 ...
6
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1answer
209 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 ...
2
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0answers
63 views

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 ...
4
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2answers
312 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), ...
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2answers
550 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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1answer
340 views

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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1answer
118 views

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(...
2
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0answers
322 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 ...
8
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0answers
192 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 ...
3
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1answer
204 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 ...
9
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1answer
331 views

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 $h\geq N+2$ ...
3
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0answers
67 views

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 $B$, $...
3
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1answer
338 views

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 $|\;\;|_p$. ...
5
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1answer
236 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 ...
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2answers
308 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 ...
4
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1answer
655 views

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 $...
2
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2answers
152 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 $\rho:G\to B(Y)$ ...
4
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1answer
809 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 ...
3
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1answer
156 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,\...
1
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1answer
106 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 ...
2
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0answers
259 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 ...
3
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1answer
230 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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3answers
2k 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 ...
9
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6answers
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 ...
2
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1answer
299 views

Extending smooth irreducible representations

Hi, Let $G_1, G_2$ be topological groups with $G_1 \subset G_2$ is closed. Let $\rho:G_1 \to Aut(V)$ be a smooth irreducible representation. Can anyone tell me if there is a criterion/example/idea ...
5
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2answers
849 views

Parabolic induction GL(n,Zp)

Let $P$ be a parabolic subgroup of $GL(n)$ with Levi decomposition $P =MN$, where $N$ is the unipotent radical. Let $\pi$ be an irreducible representation of $M(\mathbf{Z}_p)$ inflated to $P(\mathbf{...
3
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1answer
255 views

Inducing from cocompact subgroups

Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of ...
4
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1answer
259 views

Induced hermitian module

I've read about inducing representations of H modules to G modules, where H is a subgroup of G. If the H module has a hermitian form on it (hermitian with respect to the involution on k[H] sending h->...
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387 views

What is the “permanence relation” really?

I have come across the words "permanence relation" in a 1969 paper by Keith Hannabuss The Dirac equation in de Sitter space. The only other similar google hit for this phrase appears in another paper ...