1
vote
1answer
128 views

Unipotent conjugacy classes

Consider a connected reductive group G over the complex numbers. Is there a `simple' formula for the number of conjugacy classes of unipotent elements in G?
6
votes
0answers
118 views

Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...
7
votes
1answer
208 views

Open cell decomposition after applying a Weyl group element

Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step. For ...
4
votes
1answer
258 views

Does there exist a categorical treatment of root data(systems)?

What I am looking for is an abstract description of root data with their morphisms(!) plus a comparison with the categories of reductive groups over some field, Dynkin diagrams, Lie algebras, ...
1
vote
1answer
203 views

Heights in reductive groups

Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular ...
1
vote
1answer
227 views

Intertwining Integral defined on a Weyl group?

Why does the intertwining integral such as the one defined in A. W. Knapp's paper "Intertwining operators for semisimple groups" depend only on an element w of a Weyl group? ...
2
votes
1answer
212 views

Abel transform is an * isomorphism for SL(2, R)

Assume we conisder $G= SL(2, R)$, $K=SO(2)$ and $N$ the strict upper triangular matrices in $G$, $A$ diagonal matrices, and the Borel supgroup $B=NA$, $W$ Weyl group. Then we have an isomorphism of ...
2
votes
0answers
162 views

Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups

If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
4
votes
1answer
459 views

Character determines the representation?

Consider a semisimple Lie group or a $p$ adic reductive group $G$. To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?
2
votes
2answers
264 views

What is the normalization factor for $GL_n( \mathbb{Q}_p) // GL_n( \mathbb{\mathbb{Z}_p})$?

Consider $G = GL_n( \mathbb{Q}_p)$ and $K = GL_n( \mathbb{Z}_p)$, or more favorable replace $ \mathbb{Q}_p$ by a non archimedean field and $\mathbb{Z}_p$ by the ring of integers. Note that the space ...
1
vote
2answers
424 views

Normalizers of maximal compact groups?

Consider a reductive group over a local field. What is the normalizer of a maximal compact subgroup? If this is to general, what is the normalizer of $GL(n, \mathbb{Z}_p)$ in $GL(n, \mathbb{Q}_p)$, ...
1
vote
0answers
133 views

On closed abelian reductive subgroups of Real reductive groups

Hello everybody. I would first like to apologise for the basic question; I'm not expert on Lie Theory. Can someone please help me with the following questions Let $\mathrm{G}=\mathrm{K} ...