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For someone with algebraic geometry background, I would heartily recommend Procesi's Lie groups: An approach through invariants and representations. It is masterfully written, with a lot of explicit r …

If you are only interested in expanding polynomials then you can forgo Hilbert spaces and get an exact expansion into a finite linear combination. Spherically-invariant case Let $f$ be a polyno …

My recollection is that Rossmann's book on Lie groups has a detailed discussion of the exponential map and surjectivity issue. Matrix exponential map is equivariant under conjugation, $$\exp(gXg^{-1} …

"And are there any nontrivial mathematical applications of this situation?" Yes, this is a very important construction in algebraic geometry and representation theory! Algebraic geometry. The pap …

Here is my representation theorist's perspective: the key difference between representations and modules is that representations are "non-linear", whereas modules are "linear". I'll concentrate on the …

Any symplectic linear transformations in $T_xM$ is locally realizable as a Hamiltonian vector field, thus for questions 1 and 2, one can profitably use representation theory of the symplectic group. …

I assume that $\phi$ is an automorphism of $G.$ Note that if $\phi$ is inner then trivially $\rho$ and $\rho\circ\phi$ are equivalent, thus the answer depends only on the image of $\phi$ in the outer …

The answer to the general question is "no": If $\mathfrak{g}$ is solvable, by Lie's theorem its commutant $\mathfrak{g}^{\prime}=[\mathfrak{g},\mathfrak{g}]$ is represented by strictly upper triang …

This is an outgrowth of an extended comment dealing with the superficial difference between symplectic and orthogonal rotations embedded in the question. I posit that the theory for symplectic groups …

A short answer is that given a modular form $f$ for a congruence subgroup of $SL_2(\mathbb{Z}),$ one can lift $f$ to a function $F$ on the adelic group $G=GL_2(\mathbb{A})$ with certain properties tha …

As David remarked, this is nearly a hopeless task in general, but some special cases can be computed explicitly. If $\mathfrak{g}$ is a member of a skew reductive dual pair $(\mathfrak{g},\mathfrak{ …

Regular elements need not be semisimple! For example, in the Lie algebra $\frak{sl}_2$, every non-zero element is regular, with the centralizer spanned by the element itself. Among the elements of the …

There is a fruitful way of thinking about the algebra $\mathbb{C}[M_{n,m}]^{{O_n}\times O_m}$ that originates from the $(GL_n, GL_m)$-duality. Namely, $$ \mathbb{C}[M_{n,m}]=\bigoplus_{\lambda}V_ …

As Allen has indicated, there is no hope to solve this question even for $G=GL_n.$ However, there are some representations that are mild enough to be analogous to the vector (or standard) representati …

Based on comments from Eric Rowell and José Figueroa-O'Farrill, I assume that your $V$ is a half-spinor representation of $Spin_{10}$. The key issue is that your hypothetical decomposition of $S(V)$ i …