Since I am being asked the same question repeatedly and since the given answers are not quite correct, I post another answer despite the thread being so old. According to a talk by Domingo Luna ...

It is not only unusual, I would be surprised if that ever happens. A referee's task is to a evaluate a paper in the state it has been submitted. So he/she should not take into account any other ...

Since nobody did, I would like to mention the obvious: The 17 camels trick is routinely used in apportionment procedures called divisor methods. The task is to divide a number of $N\in\mathbb N$ ...

First, I do it all the time and don't really see the objections. A phrase like "In [S] it was shown..." is a good alternative to "Siegel showed, [S], that ...". Out of curiosity I did some cursory ...

For the first question I am not that pessimistic. At least there are candidates as follows: Recall that $Z$ is stably rational if there is $n\ge0$ such that $Z\times\mathbf A^n$ is rational. Now ...

This seems to be consequence of the paper Cartan, Henri: Variétés analytiques réelles et variétés analytiques complexes. Bull. Soc. Math. France 85 1957 77–99 Cartan shows more generally (see ...

That follows from a theorem of Schur saying that any polynomial $\sum_{k=0}^nc_k\frac{x^k}{k!}$ with $c_i\in\mathbf{Z}$, $c_0,c_n\in\{1,-1\}$, $n\ge 1$, is irreducible over $\mathbf{Q}$. I. Schur, ...

The wonderful compactification is always projective. One way to see is to use a theorem of Sumihiro which says that a normal $G$-variety is covered by $G$-invariant quasiprojective open subsets. Since ...

I don't know who found this presentation first but I can imagine that already Cartan knew it since it comes from a symmetric space. More precisely, $\mathfrak g=E_7$ has an involution $\theta$ whose ...

1) Counterexamples were found in the paper Brown, Francis; Schnetz, Oliver: "A $K3$ in $\phi^4$". Duke Math. J. 161 (2012), no. 10, 1817–1862. It is now the general feeling that most $\phi^4$-Feynman ...

If $G$ is connected and $T\subseteq G$ is a maximal torus then there is an involution $\theta:G\to G$ with $\theta(t)=t^{-1}$ for all $t\in T$. It has the property that as a representation $V^*$ is ...

The statement is true and well known. See e.g. Thms. 8.1.3 and 8.1.4 in Goodman-Wallach: Symmetry, representations and invariants, Springer GTM 255. In fact, much more is known. Let, more generally, $...

There are various generalizations of the Jordan-Hölder theorem. Beyond groups and groups with operators it holds for any equational theory which contains a Mal'cev operation. This means that from the ...

It is classical that, as $O(n)$-representation, $$ \text{Sym}^k(\mathbf R^n)=H^k\oplus qH^{k-2}\oplus q^2H^{k-4}\oplus\ldots $$ Here $q=x_1^2+\ldots+x_n^2$ is the quadratic form defining $O(n)$ and $H^...

Every finite dimensional real Lie algebra has the form $\mathfrak g=\mathfrak l\ltimes\mathfrak r$ where $\mathfrak l$ is semisimple and $\mathfrak r$ is solvable (Levi decomposition). The smallest ...

Let me complement Claudio's answer. There is indeed a definition of symmetric space which works for any Riemannian manifold $M$: For any point $p\in M$ there is an involutive isometry $\iota_p$ of $M$ ...

I assume $\text{char}\,\mathbf F=0$. Put $d:=b-a$. Because of $a^2-2ab+b^2=d^2-ad+da$ your equation is equivalent to $$ (*)\qquad d^{-1}a-ad^{-1}=1. $$ This precludes $\dim_{\mathbf F}D<\infty$ (...

A torus $T$ is quasi-split if its character group is a permutation representation for the Galois group. So a counterexample to your question is: let $G$ be the quasi-split group $SO(n+1,n-1)$, $n\ge2$,...

The invariants are generated by the quadratic polynomials $(u,u)$, $(u,v)$, and $(v,v)$ where $(.,.)$ is the scalar product defining $O(n)$. This pattern generalizes to arbitrary many copies of $\...

The subspace $V_\lambda$ is very easy to see. Since $U^-$ is normalized by the maximal torus $T$ there is an action of $T$ on $G/U^-$ on the right. This means that $\mathbb C[G/U^-]$ carries a ...

For fields of characteristic zero one can argue as follows: Assume first that $K$ is algebraically closed. Since semisimple subgroups of $G$ correspond bijectively to semisimple subalgebras of $\...

The idea of PseudoNeo's comment settles the converse of my statement modulo that he missed four cases. According to the theory of indecomposable modules of the Dynkin quiver $D_4$ with all arrows ...

This is true in a much more general setting. Let $X$ be any normal projective $T$-variety defines over an algebraically closed field. Then I claim that $\# X^T>\dim X$. We show this in two steps. (...

If I understand the definition correctly, a connected reductive group $H$ is an endoscopic group for a connected reductive group $G$ if its Langlands dual $H^\vee$ is a connected centralizer in $G^\...

Standard monomial theory has been extended to all classical groups by Lakshmibai, Seshadri and others in the series of papers "Geometry of $G/P$ I-IX". A very concise description of standard tableaux ...

Most maximal parabolics of $SL_{2n}$ are not $\sigma$-invariant, not even up to conjugation. So $(G/P)^\sigma$ does not make sense. The correct statement is: Let $I\subseteq\{1,\ldots,2n-1\}$ be a ...

The map $\mathbb C[\mathfrak g]^G\to\mathbb C[\mathfrak g]^P$ is an isomorphism for trivial reasons: In any quasi-affine $G$-variety, $P$ and $G$ have the same fixed points. Just look at the orbit map ...

The $G^0$-action on a coset is the same as a so-called twisted action which is pretty well understood. See, e.g., Mohrdieck, S.: Conjugacy classes of non-connected semisimple algebraic groups, ...

For a fixed point your guess is right and one doesn't need Luna's slice theorem to prove it: Let $T$ be the tangent space in $x$ and let $\mathfrak m_x\subset\mathbb C[X]$ be the maximal ideal. Then ...

The principal isotropy group is $H=SL(3)\times SL(3)$: it has the right dimension (namely 16) and occurs as an isotropy group (namely of a general element of $W$). Now it is a general result of Luna-...