Questions tagged [induced-representations]

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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 $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...
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 $1 \le i \le m$, ...
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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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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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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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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 ...
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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 ...
849 views