6
votes
1answer
358 views

Getting the story of Dynkin and Satake diagrams straight

I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...
4
votes
1answer
145 views

Decomposing tensor products of modules for the orthogonal/symplectic groups in characteristic zero

I would like to know if there is a perfect analogue of the classical Littlewood-Richardson rule for decomposing tensor products of simple modules for the orthogonal/symplectic groups in characteristic ...
2
votes
2answers
119 views

Connectedness of Springer Fibers

Let $G$ be a connected, simply-connected, complex semisimple Lie group with Lie algebra $\frak{g}$. Let $\mu:T^*\mathcal{B}\rightarrow\mathcal{N}$ be the Springer resolution of $\mathcal{N}$. If ...
5
votes
1answer
387 views

A question about the proof of Beilinson-Bernstein localisation

I'm trying to understand the proof of the Beilinson-Bernstein localisation theorem at the moment, but there's just one point where I'm having a mental block, and was wondering if anybody could clarify ...
3
votes
1answer
282 views

Centralizers of Nilpotent Elements in Semisimple Lie Algebras

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of ...
3
votes
0answers
202 views

Reductive Lie Groups and Complexification

Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, ...
1
vote
1answer
167 views

A Criterion for Reductivity of Lie Subgroups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one ...
2
votes
2answers
219 views

Stabilizers for Nilpotent Adjoint Orbits of Semisimple Groups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (ie. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...
3
votes
2answers
291 views

degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$

How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.
2
votes
3answers
353 views

Dimension of Unipotent Radicals

A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...
8
votes
1answer
271 views

On q-Demazure operators

Setup Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup $B$. Let $\Lambda$ denote the weight lattice of $G$; we write elements ...
4
votes
2answers
444 views

Failure of Jacobson Morozov in positive characteristics

The Jacobson-Morozov theorem that any nilpotent $e$ in the lie algebra of a simple algebraic group $G$ can be embedded in an $sl_2$-triple, has a restriction (in terms of the coxeter number) on the ...
5
votes
2answers
471 views

Kostant's theorem on invariant polynomials in positive characteristic

Let $k$ be an algebraically closed field and let $G$ be a reductive linear algebraic group over $k$ with Lie algebra $\mathfrak g$. If the characteristic of $k$ is $0$ then, by a classical result of ...
4
votes
0answers
130 views

The Killing form on quantized enveloping algebras and reduction to the classical case

Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
5
votes
0answers
137 views

On an interesting subalgebra of the functions on the cotangent bundle of the flag variety

Setup Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the ...
29
votes
5answers
2k views

Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
5
votes
2answers
451 views

Lie algebras and non-smoothness of centralisers in bad characteristic

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. For $x\in G$, let $C_{G}(x)$ denote the centraliser, considered as a group scheme over $k$. If ...
3
votes
0answers
144 views

Generators and relations for the enveloping algebra of a unipotent radical

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0 and let $B \subseteq G$ be a Borel subgroup with unipotent radical $U$. Let $P \supseteq B$ be a ...
3
votes
3answers
448 views

About $G$-modules versus $Lie(G)$-modules for algebraic groups

Hello, I would like to know clear references about the following facts: Let $G$ be a connected algebraic group (over alg. closed field in char. 0), $Lie(G)$ its Lie algebra, $M$ a $G$-module. I ...
10
votes
3answers
374 views

Uniform setting for computing orders of algebraic groups over finite quotients of the integers?

A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over ...
1
vote
0answers
155 views

Exotic Chains for Group Homology of a Complex Lie Group

Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural ...
4
votes
0answers
146 views

Exotic Chains for Group Cohomology of a Complex Lie Group

Related Question: Exotic Chains for Group Homology of a Complex Lie Group Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural ...
2
votes
1answer
310 views

Springer isomorphisms and parabolics

Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let ...
4
votes
0answers
186 views

How to decide if two surfaces occurring in Springer theory are isomorphic?

In the study of a simple algebraic group (say over $\mathbb{C}$) and related geometry of its flag variety associated with the Springer correspondence, one encounters pairs of surfaces which have some ...
3
votes
1answer
95 views

Lower bound on the degree of a product of elements in a hyperalgebra/enveloping algebra

Background: Fix a linear algebraic group $G$ over an algebraically closed field $k$ of arbitrary characteristic and let $B \subseteq G$ be a Borel subgroup with unipotent radical $N$. Let $\Delta^+$ ...
4
votes
0answers
297 views

about ℓ-adic and Perverse Stuff and ℓ-adic cohomology with compact support

this question is trivial. We know from this paper link text, Springer constructed rep of the Weyl group $W$ on the cohomology of the Springer fibre. Also, Deligne-Lusztig constructed the linear rep ...
6
votes
0answers
377 views

Uniform proof of dimension formula for minimal special nilpotent orbit?

Given a simple Lie algebra over an algebraically closed field of good characteristic such as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...
4
votes
2answers
1k views

Lie algebras of algebraic groups

Where can i find material about the definition of the exponential morphism from the Lie algebra of an algebraic affine group to the group?
6
votes
2answers
529 views

Relative Lie Algebra cohomology and sheaf cohomology

(I apologize in advance for the vagueness of my question). Let $G$ be a reductive algebraic group over $\mathbb C$ with Lie algebra $\frak g$ and Borel $B$. I have seen casual references to the fact ...
6
votes
0answers
287 views

Kazhdan-Lusztig graph for the Springer fiber of the minimal special unipotent class?

This graph was determined in the case of simply-laced root systems by Igor Dolgachev and Norman Goldstein: "On the Springer resolution of the minimal unipotent conjugacy class" (J. Pure Appl. Algebra ...
-1
votes
1answer
602 views

Semisimple elements of a lie algebra

Let $G\subset GL_n(\mathbb{C})$ be an algebraic group of dimension n, and let $\mathfrak{g}$ its Lie algebra.Is there a relations between the maximal number of independent semisimple elements of $G$ ...
5
votes
1answer
308 views

Differential of a nilpotent or semisimple element

Let $G$ be an algebraic connected subgroup of $GL_n(\mathbb{C})$ and let $\chi : G \to \mathbb{C}^*$ a character. Consider $d_e\chi : \mathfrak{g} \to \mathbb{C}$ the differential of $\chi$ at the ...
1
vote
1answer
362 views

on a characterization of parabolic subgroups

Over a base field $k$, linear $k$-groups stand for affine algebraic $k$-groups. For simplicity take $k$ to be a field of characteristic zero, as in this case one has the correspondence between ...
2
votes
3answers
293 views

Is the space of polynomial functions on M_n a faithful U(gl_n)-module?

We are over some field $k$ of characteristic $0$. The general linear group $\mathrm{GL}_n$ canonically acts from the left and from the right on the space $\mathrm{M}_n$, and thus also acts from the ...
2
votes
2answers
379 views

Lie Algebras and Simple Connectivity for general algebraic groups

In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always ...
11
votes
6answers
1k views

How do I stop worrying about root systems and decomposition theorems (for reductive groups)?

I apologize for this being a very very vague question. Just as personal experience, I never feel that I fully grasped the theory of root systems in Lie algebras and Lie/algebraic groups (I shall ...
2
votes
1answer
606 views

Weyl group Invariants

What are the generators of $\mathbb C[V^m]^W$, where $W$ is the Weyl group of type $E_6, E_7, E_8$, V^m denote 'm' (m > 1) copies of the Cartan subalgebra and the action is the diagonal action? Is ...
33
votes
2answers
1k views

What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped ...
8
votes
5answers
754 views

How do you switch between representations of an algebraic group and its Lie algebra?

I'm interested in the structures of categories like $Rep(GL_n), Rep(SL_n)$, etc. of algebraic representations of an algebraic group. I understand that there should be some relation between these and ...
6
votes
3answers
2k views

Simultaneous diagonalization

I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ ...
7
votes
3answers
954 views

Constructing Affine Kac-Moody Groups

Does anyone know a simple construction for Affine Kac-Moody groups? There is a book by Kumar ("Kac-Moody groups, their flag varieties, and representation theory") that does the construction for the ...
16
votes
8answers
2k views

Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?

Harold Williams, Pablo Solis, and I were chatting and the following question came up. In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...