User th.ng - MathOverflow

description of an endomorphism algebra

2012-06-15T22:21:05Z

Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).

I would like a reference for the description of the algebra $End_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd like to relate it with a structure of Hecke algebra, which is usually defined as $End_{G^F}( \mathbb{C}[G^F/B^F] ) := End_{G^F} ( Ind_{B^F}^{G^F} 1 )$. I hope to find that the endomorphism algebra is isomorphic to some kind of extension of the Hecke algebra by the torus $T$.

Thank you!

Double coset isomorphism

2012-05-30T10:58:05Z

Let $G$ be a connected reductive group (although I don't think this is relevant here), $B$ a Borel subgroup containing a maximal torus $T$ and $U$ the associated unipotent radical ($U^-$ the opposite unipotent subgroup). Finally, let $v \in W$ and $\dot{v}$ a lift of $v$ in $N_G(T)$.

How do you prove that there is an isomophism of varieties $$ (U\dot{v} \cap \dot{v}U^-) \times U \to U\dot{v}U, $$ which is the multiplication $(x,y) \mapsto xy$ ?

Thank you!

Long exact sequence of cohomology and fibration

2012-04-23T19:07:31Z

In an article I'm reading, the author is stating :

$O$ is isomorphic to the complement of a zero section of a line bundle over $X$. We have a long exact sequence of (étale) cohomology associated to the fibration : $$ \cdots \to H^{i-2}_c(X) \to H^{i}_c(X) \to H^{i+1}_c(O) \to H^{i-1}_c(X) \to \cdots $$

Is there a result involving a long exact sequence (of cohomology) and the complement of a zero section of a (line) bundle ?

For now, I'm guessing the first and the last term of the sequence above are the cohomology groups of the line bundle : if $Y \to X$ is the linde bundle, then $$H^i_c(Y) = H^{i-2}_c(X).$$ So hypothetically there should be a long exact sequence like : $$ \cdots \to H^{i}_c(Y) \to H^{i}_c(X) \to H^{i+1}_c(O) \to H^{i+1}_c(Y) \to \cdots $$ but I don't understand why.

Thank you.

Description of $GL_3/U$

2012-01-31T17:01:19Z

Let $U$ be the set of unipotent upper triangular matrices and $B$ the upper triangular matrices of $GL_3$. How could I describe $GL_3/U$ ? Using coordinates, in a projective or an affine space.

For example, I already know the identification of $SL_2/U$ with $\mathbb{A}^2 \setminus (0,0)$ and the identification of $SL_2/B$ with $\mathbb{P}^1$ ($B,U$ the standard Borel of $SL_2$ and its unipotent radical).

Thank you.

Longest element of a Weyl group

2011-09-12T15:47:28Z

Let $G$ an algebraic (reductive) group. $T$ a maximal torus, $B$ a Borel subgroup containing $T$, and $w_0$ the longest element of the Weyl group.

I'm looking for a reference explaining why when you conjugate $B$ by $w_0$, the result is the opposite Borel subgroup $B^-$.

Is there a proof involving roots of $G$ relative to $T$ ?

I've found a proof in a book of M. Geck, but this proof doesn't involve roots at all, but only the fact that $B^-$ is uniquely defined by the relation $B \cap B^- = T$.

Frobenius eigenvalues and Deligne-Lusztig theory

2011-06-23T15:05:27Z

In the article of Lusztig (1977) I'm reading, about representation theory of finite Chevalley groups, after introducing unipotent representations, and the Deligne-Lusztig Variety $X_w$, there are some results related to the number of rational points on a variety and eigensvalues/eigenspaces of $F^\delta$, with $\delta$ chosen such that $F^\delta$ acts trivially on $W$.

Is there any reference explaining how the eigenspaces/eigenvalues are related to the study of unipotent representation ? Because to me, it seems to appear out of nowhere, and I'm not sure why the author is doing that.

Maximal torus and parabolic subgroups in reductive groups over finite fields

2011-06-05T17:29:30Z

Let $G$ be a reductive group, $B_0$ a $F$-stable Borel subgroup and $T_0$ a fixed $F$-stable maximal torus contained in $B_0$. ($F$ = Frobenius morphism). Let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$.

For $w$ in the Weyl group associated with $T_0$, let $x \in G$ such that $x^{-1} F(x) = w$ and define $T_w = x T_0 x^{-1}$.

It is then stated in the article I'm reading that :

If the maximal torus $T_w$ is not contained in any proper $F$-stable parabolic subgroup of $G$, it is well known that this implies $l(w) = r\ ( mod\ 2 )$.

Does anyone has a reference for this ?

Edit (following Jim's advices) : The article I'm currently working on is "Representations of finite Chevalley groups" from Lusztig (1977) (CBMS Regional Conf. Series in Math.n°39). Even more specifically, I'm trying to understand many examples (3.10) that are not fully developped about unipotent representations. The previous quote is from the proof of the following claim :

Assume $q$ is greater than the Coxeter number of $G$. Let $\rho$ be an irreducible cuspidal $G^F$-submodule of $H^{i}_{c}(X_w)_{\mu}$ and let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$. Then all complex conjugates of $\mu$ have absolute value of the form $q^{\delta m /2}$ where $m$ is an integer congruent to $r$ modulo $2$.

Answer by th.ng for Maximal torus and parabolic subgroups in reductive groups over finite fields

2011-06-06T16:28:03Z

Thank you for adding more to the general context, pointing out what is important. As a matter of fact, you are completely right, the Borel $B_0$ is chosen $F$-stable. (I edited the post).

Closer to the question, since I met this morning my advisor (for my minor thesis) he sketched me a proof for what, however, did not seem to him as "well known". Just a warning though, the proof is not clear at the moment for me, but I'll try to state it anyway (for those interested by the answer and able to decipher what I'll be writing).

The case where $T_0$ is split is easier. One has to start with the claim that the condition for $T_w$ not to be contained in any $F$-stable parabolic subgroups implies that $1$ is not an eigenvalue for $w$ acting on $X(T)$. Then, as the eigenvalues for $w$ (in general) are $1$, $-1$ or pairs of conjugate complex numbers. So, by looking at the determinant of $w$, one should be able to conclude.

Special case of Leray spectral sequence

2011-05-31T14:40:45Z

I am looking for a reference for what is stated in Srinivasan's book "Representations of Finite Chevalley Groups", which is apparently a special case of Leray spectral sequence. I'll quote the interesting part :

Let $\phi$ be a torsion sheaf on $X$, $f : X \to Y$ a morphism of schemes. Then we have a Leray spectral sequence $ E_2^{pq} = H^p_c(Y, R^q f_{!} \phi ) \Rightarrow H^{p+q}_c(X, \phi)$. The way we use this is as follows.

Suppose that all the fibres of $f$ are isomorphic to a fixed scheme $Z$ such that $H^q_c(Z, \phi) = 0$ except for $q= q_0$. Then $H^p_c(Y, R^{q_0} f_{!} \phi ) \simeq H^{p+q}_c(X, \phi)$.

The reference I'm looking for is for the second part of what I quoted.

Since I just happened to work with spectral sequences, maybe this statement is obviously equivalent to the definition of convergence of spectral sequences, in which case I'm sorry for asking.

Borel subgroups contained in a fixed parabolic subgroup

2011-05-27T11:55:23Z

The question is asked in the context of (connected) reductive groups.

In the article i'm working on, the author states the following fact (well it's not word to word exact, I simplified it a little) :

If we choose a parabolic subgroup, determined by a simple reflection $s$ in $W$ (the Weyl group, given a maximal torus), then the variety of the Borel subgroups contained in $P$ is one-dimensional, moreover it can be identified with the projective line.

Is there a simple way to prove this ?

What I tried : since the parabolic subgroup $P$ is determined by a simple reflection, I wrote down the Levi decomposition using roots, then I was thinking that the Borel subgroups contained in $P$ are in bijection with the Borel subgroups of the Levi complement.

Comment by th.ng

2012-06-25T08:11:57Z

Now that I've read what you mention, the generators/relations presentation of the Yokonuma-Hecke algebra is (almost) precisely what I needed. Though, I'll also have to see how what Jim said could fit in. Thank you again.

Comment by th.ng

2012-06-21T07:44:32Z

Thank you for this reference, I'll read i too, it could be useful too.

Comment by th.ng

2012-06-20T08:33:46Z

Thank you for your answer, actually, I didn't expect the permutation module $\mathbb{C}[G^F/U^F]$ so be the direct sum of all the induced representations, can you tell me why ? or give me a reference for it ? At the moment, what I want to do isn't clear : I was looking for something to replace the (usual) Hecke algebra, which is related to the unipotent representations of $G^F$, in the case where one considers the other representations (all this is related to Deligne-Lusztig theory and more precisely, instead of studying the varieties $X_w$, I want to have something for $Y_w$).

Comment by th.ng

2012-05-30T16:02:45Z

I really meant $U\dot{v}U$, but I think what you mention will do the trick. I will read it then :) thank you!

Comment by th.ng

2012-04-24T11:39:28Z

Indeed I thought about that later. Anyway, thank you very much for your help!

Comment by th.ng

2012-04-24T06:47:09Z

Thank you Dan. Though I already knew about the "open-close" long exact sequence, I wasn't sure how it fits in my settings. Maybe I don't understand well what is the meaning of "complement of the zero section of a line bundle"...

Comment by th.ng

2012-03-26T05:16:29Z

You can read $SL_2(\mathbb{F}_q)$ from C.Bonnafé, there's a description of the non-split torus in this case.

Comment by th.ng

2012-01-31T20:23:58Z

I don't understand the first identification of $SL_3/U$, could explain it ? Moreover, do you have some kind of references for the generalization to $GL_n$ or any $G$ ?

Comment by th.ng

2012-01-31T18:37:38Z

Actually, I was looking for a more explicit identification, for example, I was able to embed $GL_3/B$ in $\mathbb{P}^2 \times \mathbb{P}^2$, so I was looking for something like that. But I'll see if I can get something from the Bruhat decomposition. Thank you.

Comment by th.ng

2012-01-31T18:34:40Z

You're both right, I knew it is a projective variety, but as I'm trying to work out a proof in the setting of G= GL_3, I'm willing to explicit the construction with coordinates.

Comment by th.ng

2011-09-12T20:21:12Z

Indeed, I worked with those 3 references and especially with yours, and I was a wondering if I missed something about this longest element or not. Though you are right, with 26.3(b) I thought I was really next to a proof, but still, as I commented above, one has to examine precisely the action of $w_0$, and the fact that $w_0$ send $\phi^+$ on $-\phi^+$ does not seem obvious to me, without using the action on Weyl chambers.

Comment by th.ng

2011-09-12T19:21:09Z

Thank you ! I was trying to do the same thing but on the group-level, meaning that I was trying to look at the $U_\alpha$ (whose Lie algebras are the $\mathfrak{g}_\alpha$). However, I was trying to explicitely determine the image $w_0(\alpha)$ and I couldn't figure it out... I'll look at the global image of the positive roots then.

Comment by th.ng

2011-09-12T19:16:54Z

Thank you, indeed, I've seen a proof that a Weyl chamber is sent to the opposite under the action of the longest element using combinatorics, but that wasn't really my setting... I should have been more precise.

Comment by th.ng

2011-06-06T06:43:59Z

I edited my post, but concerning the fact that the fixed torus is probably of maximal split rank, I can't say it is in the setting of proof... but maybe I'm wrong.

Comment by th.ng

2011-05-27T20:49:43Z

Thanks, I edited it.