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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(\...

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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 ...

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### 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{...

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### 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 ...

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### 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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### 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 ...