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Yes - this statement is true for any irreducible representation of a split reductive p-adic group $G$, where $U$ is the unipotent radical of the Borel (a Borel subgroup is a minimal algebraic subgroup …

Decided to add a new answer instead of editing the old one (which answers a different question). Look at the proof in Bernstein-Zelevinsky. It generalizes to arbitrary split reductive groups. Here's a …

Suppose you are given a linear combination of points of $\mathbb{Z}^n$ which corresponds to the weight datum of a nontrivial representation $V$ of some simple Lie algebra $g$. Is it possible to recons …

I'll write this as an answer rather than a comment to close the question. Victor Protsak's comment gives the original question a definitive answer of no. The formulation of the original question is …

Say $\mathcal{C}$ is the Abelian category of finitely-generated modules over some $k$-algebra $A$. Then an object $M\in \mathcal{C}$ is finite-dimensional over $k$ if and only if $\text{Hom}(P, M)$ is …

The Pauli spin vector encodes a two-complex-dimensional unitary representation of the Lie algebra $\bf{su}_2$. A basis of the Lie algebra maps to the matrix algebra $\bf{gl}_2(\mathbb{C}),$ giving a $ …

Sorry if this would be more appropriate as a stackoverflow and not a mathoverflow question, but I think it's more likely to be known in this community. There are plenty of places on the internet or i …

If you introduce super-structure, it doesn't matter whether $\mathbb{C}(1/2)$ is even or odd. The reason is that your category of motives has a symmetric monoidal automorphism which acts by $-1$ on $\ …

For a partition $\lambda=(\lambda_1\ge \lambda_2\ge \dots)$, let $m_\lambda=\prod_i (\lambda_i-\lambda_{i+1})!$ be the product of factorials of consecutive differences and let $v_\lambda=\prod_{i | \ …

Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})_\lambda$ be the $\lambda$'th …

This question popped up on the feed recently, and I wanted to add another, "brave new math"-style answer. Namely, the point of Lie algebras is that, in characteristic $0$, a Lie algebra (resp., an $L_ …

The short answer is that the characteristic $p$ picture genuinely has more depth. In particular it's not correct to think of formal groups as "better" than Lie groups. In fact, they can be put on equa …

Unless I'm mistaken you can say "coalgebra with conilpotent coabelianization" (the Hopf algebra structure doesn't seem relevant here). Namely, first note that any one-dimensional representation of a c …