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I agree with David Speyer's answer, and furthermore there is no canonical way to construct $V_i$ from $V$. This is a subtle and oft-overlooked point in representation theory, in my opinion. Many tex …

As I understand the question, the OP would be happy to see a description of the lowest-dimension fundamental representation of $F_4$ (and perhaps $E_6$, $E_7$, $E_8$), and is happy with the descriptio …

I'd recommend first that you and your friend spend more time with Tits :), "Reductive groups over local fields", from the Corvallis volume (free online, last time I checked). Undoubtably there are ot …

I don't know anything about quantum groups, but the Peter-Weyl theorem for compact groups generalizes nicely to Type I second-countable locally compact topological groups, a result of Segal and Mautne …

I think, although it's dated later than Deligne's paper that you mentioned, that the first written instance of Kazhdan's principle is in the paper "Representations of groups over close local fields", …

This is a common point of confusion, and the OP is on exactly the right track. A good reference for the representation theory is Chapter 1, Section 6, of Jacquet-Langlands book "Automorphic forms on …

I think it's best to look at the relatively recent paper of Moshe Baruch, Annals of Math., "A Proof of Kirillov's Conjecture" -- in the introduction of his paper, he discusses the basic techniques of …

I'm writing up some notes, and I realize I don't have a counterexample for something I suspect is false. Dubious claim: If $(\pi, V)$ and $(\rho, W)$ are irreducible representations of two groups $G …

First a few background references: Read Mackey's books or articles for much more on unitary representations. Especially the Bull. AMS article "Infinite-dimensional group representations" from 1963 i …

I think the best way to see the signature of these quadratic forms is by using the formula from "A Classification Theorem for Albert Algebras" by R. Parimala, R. Sridharan, and Maneesh L. Thakur, Tran …

Nope. Really, this is a linear algebra question. You can take tensor products of pairs of vector spaces, symmetric and exterior powers of a single vector space. These are all functorial, so extend …

Though Peter's answer addresses the finite-dimensional representation theory, I believe that the question asks about the unitary representations on Hilbert spaces, and more general irreps on Banach an …

Try using Frobenius reciprocity. Let $V$ and $W$ be two representations of $H$, and let $U$ be a representation of $G$. Consider first the space: $$Hom_G \left(U, Ind_H^G (V \otimes W) \right) \con …

The general conjectural picture is the Gross-Prasad conjecture, found in Section 2 of Gross and Prasad's paper "On the decomposition of a representation of $SO_n$ when restricted to $SO_{n−1}$." The …

This question was treated by Monica Nevins in the following pair of papers. Nevins, Monica, Branching rules for principal series representations of SL(2) over a $p$-adic field, Can. J. Math. 57, No …