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13 votes
0 answers
797 views

Borel-Weil-Bott, Langlands and Hitchin

Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote by $B$ a Borel subgroup of $G_\mathbb{C}$. Given a dominant weight $\lambda$, one can construct a line ...
Satoshi  Nawata's user avatar
6 votes
0 answers
173 views

Equivalence between $\mathcal{D}_\lambda$ modules and $\mathcal{D}_{0}$ modules

Fix $G$ a finite dimensional reductive group and $\lambda$ a weight. Apparently the category of $\mathcal{D}_\lambda$ modules on $G/B$ is equivalent to the category of $\mathcal{D}_0$ modules on $G'/B'...
Pulcinella's user avatar
  • 5,701
5 votes
0 answers
324 views

A question about equivariant derived categories and [BBD]

Let $G$ be an algebraic group (over $\mathbb{C}$) acting algebraically on a variety $X$. Bernstein and Lunts then define in [BL94] the equivariant derived category $D^b_G(X,\mathbb{C})$ (of $\mathbb{C}...
Oliver Straser's user avatar
4 votes
0 answers
99 views

Confusion about twisted Vermas in Feigin-Frenkel

Let $G$ be a finite dimensional semisimple algebraic group, and for $s\in W$ write $i_s: F_s=B^+sB^-/B^-\to F$ for the $s$th Bruhat cell in the flag variety $F$. Then (in Affine Kac-Moody Algebras and ...
Pulcinella's user avatar
  • 5,701
3 votes
0 answers
119 views

Different definitions of the thick affine flag variety

I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same. Some ...
Qixian Zhao's user avatar
2 votes
0 answers
97 views

An injective map in equivariant algebraic K-theory

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent ...
fool rabbit's user avatar
2 votes
0 answers
129 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
fool rabbit's user avatar
2 votes
0 answers
640 views

Areas of algebraic geometry useful for geometric representation theory

What topics/areas of algebraic geometry (aside from perverse sheaves/D-modules, etale cohomology, and possibly derived algebraic geometry) is it useful to learn/master if one is interested in doing ...
Yellow Pig's user avatar
  • 2,964
2 votes
0 answers
150 views

Projection of conormal bundle of Schubert variety under Springer resolution

Let $G=\mathrm{GL}_n(\mathbb{C})$ and $X_{\omega}=\overline{B_-wB/B}\subset G/B$ be a Schubert variety. Denote by $C(X_\omega)$ the conormal variety inside $T^*(G/B)$ , $\mu:T^*(G/B)\to \mathcal{N}$ ...
Ben's user avatar
  • 849
2 votes
0 answers
228 views

References for crystal bases and Demazure modules in representation theory

I was wondering what are some standard general references/books/survey articles about: (1) crystal bases, and string parameterizations and (2) Demazure modules, and Schubert varieties (containing ...
Kiu's user avatar
  • 893
1 vote
0 answers
101 views

Characteristic functions of character sheaves on tori

I am currently reading a set of lecture notes by V. Ostrik and G. Williamson, Character sheaves, tensor categories and non-Abelian Fourier transform. In Theorem 1.1, they make the assertion that the ...
Martin Skilleter's user avatar