Questions tagged [induced-representations]
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54
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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 ...
2
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1
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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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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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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
$$
(...
- 593
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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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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 ...
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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), ...
- 6,040
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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$ ...
user91659
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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)$ ...
- 7,266
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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 ...
- 360
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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{...
user36116
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3
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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 ...
- 22.8k
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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 ...
- 22.8k
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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 ...
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