# Questions tagged [induced-representations]

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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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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 $\... 4 votes 2 answers 404 views ### A morphism intertwining two induced representations TL;DR: Given representations$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 $$(... 4 votes 0 answers 321 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 votes 0 answers 149 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 ... 1 vote 0 answers 97 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 ... 3 votes 0 answers 104 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 ... 2 votes 1 answer 354 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 votes 0 answers 203 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) ... 3 votes 0 answers 52 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 ... 8 votes 1 answer 762 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 votes 0 answers 86 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 votes 2 answers 624 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), ... 9 votes 2 answers 675 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 ... 16 votes 1 answer 457 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 ... 2 votes 1 answer 128 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 votes 0 answers 352 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 votes 0 answers 245 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 votes 1 answer 296 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 ... 8 votes 1 answer 348 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 votes 0 answers 75 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, ... 4 votes 1 answer 804 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 votes 1 answer 418 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 ... 3 votes 2 answers 539 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 ... 5 votes 1 answer 1k 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 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) ... 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 ... 3 votes 1 answer 200 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 vote 1 answer 122 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 ... 3 votes 0 answers 378 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 votes 1 answer 281 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{... 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 ...
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 ...