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What is the highest weight of the representation of special orthogonal group $SO(n)$ on the space of harmonic polynomials $\mathcal H_m(\mathbb R^n)$?

In addition to @Vít Tuček's answer, here is a geometric way to explain the weights of $\mathcal H_m$ and their multiplicities. We fix $G=SO(2n)$. Let $\varepsilon_1,\dots,\varepsilon_n$ be the ...
emiliocba's user avatar
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6 votes

What is the highest weight of the representation of special orthogonal group $SO(n)$ on the space of harmonic polynomials $\mathcal H_m(\mathbb R^n)$?

This can be found in many places usually under the name spherical harmonics. See e.g. section 5.6.4 of Symmetry, Representations, and Invariants by Goodman and Wallach. The highest weight of this ...
Vít Tuček's user avatar
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0 votes

Orthosymplectic superalgebra

It is explained in Page 32 in Lie Superalgebras.
Jack's user avatar
  • 417
5 votes
Accepted

Restriction of scalar commutes with taking maximal subtorus for semisimple group G

One can test whether a subgroup is a maximal torus (as @YCor commented, not the maximal torus, unless you're in a commutative group) after base change, so it suffices to observe that the isomorphism $\...
LSpice's user avatar
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1 vote

A correspondence between projective representations of $G$ with those of its universal cover

Let $G=S^1$ which has a universal cover $\mathbb R\to S^1:t\mapsto e^{2\pi it}$. Now $\mathcal S\colon \mathbb R\to U(\mathbb C^2)/U(1)$ given by $t\mapsto \begin{pmatrix}e^{\pi i t}\\&1\end{...
Kenta Suzuki's user avatar
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5 votes
Accepted

Iwasawa decomposition of a non-compact semisimple Lie group?

For simplicity, I will work only with connected Lie groups (and, accordingly, identity components of isometry groups): Real hyperbolic space $M=\mathbb H^n$: $G=SO^+(n,1)$: $K=SO(n)$, $N\cong {\...
Moishe Kohan's user avatar
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6 votes
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Is the exponential map of a locally compact group a local homeomorphism?

No. Suppose that $G=(\mathbf{R}/\mathbf{Z})^\mathbf{N}$. Then $\mathrm{Hom}(\mathbf{R},G)=\mathbf{R}^\mathbf{N}$ is not locally compact, hence cannot be locally homeomorphic to $G_0=G$.
YCor's user avatar
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0 votes

Explicit formula for complex structure on flag manifold/isospectral matrices?

It's a bit unclear to me what conventions you're using to identify $\mathfrak{u}(n)^*$ with $Herm(n)$ - it seems like you're taking $\langle X, Y\rangle = i\operatorname{Tr}(XY)$ or something. But I'm ...
user17945's user avatar
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