15
votes
Accepted
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$ as $\operatorname{GL}(n,\mathbb{C})$-modules
This is a problem about plethysm. Example 9 from Macdonald's book, in the section on plethysm, gives the formulas
$$ e_2 \circ e_k = \sum_{j \text{ odd}} s_{(k+j,k-j)'}$$
and
$$ h_2 \circ e_k = \sum_{...
- 40.2k
7
votes
Accepted
Ergodicity of action of finite index subgroups in the boundary
Let $X$ be a Riemann surface of class $P_G$ (i.e. which carries a Green function) but is Liouville (i.e. admits no nonconstant bounded harmonic functions). One way to construct these is to take a $\...
- 12.2k
7
votes
Accepted
Lie subalgebra annihilated by all derivations
A class of counterexamples is given by the so-called characteristically nilpotent Lie algebras. For a Lie algebra $\mathfrak{g}$, consider the central descending chain defined recursively by
$$\...
- 5,878
5
votes
Centralizers in semisimple Lie group
The condition is known as "strongly regular" (and I think that the modern usage is "regular semisimple", hence also "strongly regular semisimple", not just "regular&...
- 12.9k
5
votes
Accepted
Explicit Jacquet-Langlands correspondence for real reductive groups
The short answer to all of your questions is: Jacquet-Langlands for $GL(n,F)$ is special because $L$-packets for $GL(n,F)$ are singletons. Consequently the kinds of things you are asking about for ...
- 761
3
votes
Which compact Lie groups have an upper bound on the dimension of irreducible continuous representations?
Compact Lie groups have nice classification, see e.g. https://en.wikipedia.org/wiki/Compact_group#Compact_Lie_groups and references therein.
Theorem: Every connected compact Lie group is the quotient ...
- 8,597
3
votes
Stabilizers of the action of Levi on abelianization of nilpotent radical
I am posting this as an answer since it is too long for a comment.
Consider the case that $G$ equals $\textbf{SL}(W)$ for a finite dimensional vector space $W$. Let $W=S\oplus Q$ be a direct sum ...
2
votes
Accepted
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I believe that the answer to your question is yes and it follows from Proposition 8.1 in our paper The Mirković–Vilonen basis and Duistermaat–Heckman measures (with Baumann and Knutson). The notation ...
- 4,612
2
votes
Online References for Cartan Geometry
The videos of the Training School on Cartan Geometry, held in Brno, Czechia, September 4-8, 2023 are here on YouTube. They include talks on:
Boris Doubrov (Minsk), Cartan geometry via exterior ...
- 26.3k
2
votes
Stabilizers of the action of Levi on abelianization of nilpotent radical
This is an example of a "internal Chevalley module," which has been studied in a bunch of contexts. A good reference is ``On the structure of parabolic subgroups of algebraic groups'' by ...
- 2,516
2
votes
Stabilizers of the action of Levi on abelianization of nilpotent radical
I don't know the answer to your questions in this generality, but let me mention a rich class of examples. Let $P$ be the maximal parabolic corresponding to the highest root of $G$, and let $V = V_U$ ...
- 5,760
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