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Let $R$ be a commutative ring, with whatever hypotheses let you answer the question -- e.g. Noetherian, local, finitely generated over $\mathbb C$. Let $I$ be the ideal defining the singular locus in …

My bashful, nameless, colleague asked me: When you identify opposite faces of a square, then depending on where you twist or not, you get a torus, Klein bottle, or projective plane. What spaces c …

Let $G$ be a complex reductive Lie group, $B$ a Borel subgroup, with which to define "dominant weight". Let $\lambda$ be an integral weight, not necessarily dominant, but nonetheless giving a one-dime …

Let $R$ be a Poisson algebra (over $\mathbb C$, say) with Poisson center $Z = \{c \in R : \{c,R\} = 0\}$ and consider two types of ideals $I \leq R$: $I = \langle (c_i) \rangle$ is generated by Casi …

Use Borel's theorem (about the existence of fixed points for actions of solvable groups) on the relevant Hilbert scheme, to degenerate $X_1$ to being $B$-invariant and $X_2$ to being $B_-$-invariant. …

Zhelobenko has books on the subject from 1970, 1983, 1994, 2004. I'm pretty sure it's the 1970 that I saw the most concrete branching laws in. The special cases you're interested in are really easy, …

Let's pull back to irreps of $GL_1({\mathbb R}) \times SL_n({\mathbb R})$. By Schur's lemma, the first factor acts by scalars, so the representation is of the form $V \otimes W$ where $V$ is a chara …

Hello again. Yes, it's true. The more general statement you want is, let $X$ be projective with a $K$-equivariant ample line bundle ${\mathcal O}(1)$. For each $n$, let $\mu_n$ be a measure on ${\ma …

I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles i …

This paper showed two century-old classification results, each of very undergrad-comprehensible things, were the same; pretty amazing! http://arxiv.org/abs/1308.0751 "Sums of squares and varieties o …

The general statement to know is, if $\mathbb G_m$ acts on a smooth complete scheme $X$, then each Białynicki-Birula stratum is an affine bundle over its fixed-point set. This is in the original paper …

A partial permutation matrix $\pi$ is one with at most one 1 in any row and column (the rest 0s). Given one, we can cross out to the East and South (but not Southeast) of each 1. Some boxes get crosse …

Let $H$ be a hyperplane in $\mathbb P^n$, and $X \subseteq H$ be a subscheme. Let $CX \subseteq \mathbb P^n$ be the cone on $X$ from a point $p \notin H$. Let $Hilb_{CX}$ be the Hilbert scheme whose …

On the last page of Schmid's article "Discrete Series", he says "In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure o …

Let $P$ be the convex hull of a finite set of points in $\mathbb Z^d$, and $p(n) = \#\{nP \cap \mathbb Z^d\}$ be its Ehrhart polynomial, which is also the Hilbert polynomial of the corresponding proje …