15

To follow up on what Tobias and Chuck wrote, I think you should reconsider category $\mathcal{O}$ or better yet, its principal block $\mathcal{O}_0$, but with the proviso that you have to make some sacrifices. One sacrifice is that you have to work in the derived category, not the abelian, and the other sacrifice is that you must accept a braid group action,...


14

Sometimes when you define a group using an arbitrary choice of object and then show the choice of object doesn't matter, you could have defined a groupoid without making an arbitrary choice. For example, to define the fundamental group $\pi_1(X,x)$ of a path-connected space $X$ we need to choose a basepoint $x \in X$, but then we can show we get ...


13

Here's the "answer" that I started writing, then put away for a while. The short answer is: although "T//W" is not the same as G, they "look sort of the same" from the point of view of certain generalized cohomology theories. If you replace the role of the maximal torus with "arbitrary abelian subgroups", this "looking the same" can be given homotopical ...


11

If you type "help" immediately on entering the program, you'll get a fairly long and useful introductory message. At whatever level you are, you are supposed to be able to type "help," and then the name of any command accessible at that level, to get a message about it. (Well, not all these help files exist.) The help files exist as text files in the ...


8

Let $B$ be a Borel subgroup containing $T$. As $F(B)$ and $B$ are both Borel subgroups containing $T$ there exists an element $n \in N_G(T)$ such that ${}^nF(B) = B$. Thus the Frobenius endomorphism $F' : G \to G$ defined by $F'(g) = nF(g)n^{-1}$ induces an automorphism $F' : W \to W$, where $W = N_G(T)/T$, and this stabilises the Coxeter generators $\mathbb{...


7

Let me rephrase your question. Fix a representation $\rho: G \to GL(V)$ of the Chevalley group $G$. You want a criterion for whether there is an element $x \in GL(V)$ so that $\rho(\sigma(g)) = x\rho(g)x^{-1}$ for all $g \in G$. In your example of $G = SO_{2n}$ and $V$ the natural representation, you can take $\sigma$ to be an outer automorphism of $SO_{...


7

It's important to emphasize that the simple group here is typically isomorphic to the rotation subgroup of $W$, which has index 2 and doesn't contain the reflections. So you need to look at the precise descriptions of the Weyl groups to see where the reflections fit in. There is a lot of information in the Atlas of Finite Groups (Oxford, 1985) if you ...


7

This is only a slight modification of the argument already given, but I liked it enough to type it in. Since $W$ acts with no stabilizer on the open Weyl chamber $K$, for $K$ to contain a root $\beta$, this $\beta$ cannot be perpendicular to any other root. In particular the Dynkin diagram $D$ has to be a complete graph (but the triangle $K_3 = \widehat ...


7

You are asking a number of related questions here, most of which require more reading of Lusztig's papers. See the reference list in my conference paper here, for example. But note first that the notions about cells you mention are defined quite generally for Coxeter groups in the landmark 1979 paper by Kazhdan and Lusztig here. What is special about ...


7

The first statement amounts to determining whether $T^w$ is connected. In the simplest case $G={\rm SL}_2$, $T$ may be identified with the multiplicative group ${\rm G}_m=\Bbb{C}^{\times}$, the Weyl group is cyclic of order 2 and the simple reflection $s$ acts by $x\mapsto x^{-1}$. Therefore, its fixed point set is $\{x: x^2=1\}=\{1,-1\}$, which is ...


7

The answer is "yes" and there is a geometric explanation. Let $\mathcal{H}$ denote the set of hyperplanes corresponding to the reflections $s_\alpha$ with $\alpha\in\Phi^+$ (note that $s_\alpha=s_{-\alpha}$), and let $\Sigma$ denote the connected components of $V\setminus\bigcup_{H\in\mathcal{H}}H$ (where $V$ is the vector space where $\Phi$ lives). The ...


7

This paper of Humphreys addresses your second question (the first is answered in the comments - the $W$-module structure is independent of the choice of torus): https://people.math.umass.edu/~jeh/pub/zero.pdf Here is a quote from the paper (section 1.4): Indeed, it is usually unclear how to determine directly whether or not the W-module $L_\lambda(0)$ [i.e. ...


6

This is true if $G$ is split. First of all $w$ is represented by some $\tilde w\in N_G(T)\cap G(k)$ (see Borel-Tits, for example). Then $\text{Ad}\, \tilde w$ acts on $\mathfrak g_\alpha$ by some scalar $c\in k^*$. If there is $t\in T(k)$ with $\alpha(t)=c$ then $\dot w=\tilde wt^{-1}$ will do the trick. Now if $\alpha$ is primitive in the weight lattice $\...


6

I think you can use minimal coset representatives for this. Each $w \in W(\Delta)$ can be uniquely written as $w_K w^K$ where $w_K \in W(\Delta_K)$ and $l(w_K) + l(w^K) = l(w)$. The element $w^K$ is called minimal left coset representative (and all of this works also for the opposite side) and the set of these is usually denoted by $W^K$. Now the Bruhat ...


6

Expanded with lots of references at @JimHumphreys's request. Borel is Borel - Linear algebraic groups (2nd edition). Springer is Springer - Linear algebraic groups (2nd edition). Let $G$ be reductive. A maximal torus $T$ of $G$ is connected soluble, hence contained in a maximal such subgroup $B$, i.e., a Borel subgroup; but, since all Borel subgroups are ...


5

By part (b) of Remark 1.6 on page 7 of http://www.math.lsa.umich.edu/~jrs/papers/affine.ps.gz, the answer seems to be: "Not always"! A shorter counterexample is $W=S_4$ the symmetric group on 4 letters and $I=J=\{ (12), (34) \}$. Then $\# W_I=4$ so that $W^I = \left\{ w \in S_4:\, w(1) < w(2),\, w(3) < w(4) \right\}$ has $\# W^I= \# W/W_I = 3! = 6$ ...


5

It would help to recall Dynkin's definition of "regular semisimple subalgebra" of a complex semisimple Lie algebra (say $\mathfrak{g}$), which has not become standard in later literature. This just means a semisimple subalgebra of $\mathfrak{g}$ which contains a Cartan subalgebra (sometimes called a "maximal toral subalgebra" in this special case, any two ...


5

Yes, that's true. The standard recursive constructions will give you this fact easily, because the only group elements involved in $P_{x,w}^I$ are those which are $\leq w$ w.r.t. the Bruhat order. If $w\in W_J$, then all those elements are themselves contained in $W_J$.


5

Yes, this follows from the fact that $x \mapsto w_l x$ is an antiautomorphism of the Bruhat order on a finite Coxeter group. (See Björner and Brenti, Proposition 2.3.4, for example, but their $w_0$ is your $w_l$) You also need the fact that $w_l$ is an involution. (For example, Björner and Brenti, Proposition 2.3.2.) If $w\geq w_0$, then $w_lw\leq w_{l,\...


5

Theorem 33 in the preprint [1] gives the uniqueness of BGG resolutions (= direct sums of Verma modules resolving a simple module) in category $\mathcal{O}$, both in regular and singular blocks. (However, this does not apply directly to parabolic versions of category $\mathcal{O}$). [1]: Mazorchuk-Mrđen: BGG complexes in singular blocks of category $\mathcal{...


5

Around the time Fomin and I wrote this paper, Tao Kai Lam applied the technique to type $D_n$. It emerged that it was "natural" to weight a reduced decomposition $\rho$ by $2^{d(\rho)}$, where $d(\rho)$ is the number of simple reflections in $\rho$ that correspond to the $n-2$ "nonbranch nodes" in the Coxeter diagram for $D_n$. Using this ...


5

This is easy to do in SageMath. E.g. the following code G = WeylGroup("F4") w = G.long_element_hardcoded() print(w) rw = w.reduced_words() len(rw) outputs 2144892. If you want to look at some of these reduced words just examine the list rw. To create a list for classical types of different rank do res = {} for n in range(2,5): G = WeylGroup([&...


4

I'm not sure whether there is an efficient way to answer your question (or a written reference), but it's possible to analyze the situation case-by-case. It's probably best to start with an irreducible root system $\Phi$ (reduced in the Bourbaki sense), which can be asociated with a simple Lie algebra over $\mathbb{C}$ but can just as well be studied ...


4

Apparently this isn't discussed in any of the published literature, even in the numerous exercises for Bourbaki's Chapter VI on root systems in Lie Groups and Lie Algebras. I'm not sure how strong the evidence is for the assertion here that the minimal length is always $h^\vee -2$. Even though it's true for some small rank cases, has it actually been ...


4

EDIT II: Sorry to bump this again but the answer to this question can be phrased entirely in terms of finite Coxeter groups and doesn’t depend at all on the fact that we’re dealing with $\textsf{E}_6$, so it seemed best to write the answer that way. Let $V$ be a finite dimensional real euclidean vector space and let $\Phi \subseteq V$ be a root system in ...


4

After reading your comments, I come up with the following proof. I would like to know whether my proof is correct or not. For weyl group $W$, each $w\in W$ can be expressed as $w=s_{\beta_l}\cdots s_{\beta_{2}} s_{\beta_1}$ for some distinct positive roots $\{\beta_1, \beta_2, \cdots, \beta_l\}\subseteq \Phi^+$. My proof: Let $w=s_{i_1}s_{i_2}\cdots s_{...


4

This is definitely not true. For instance, already in $\Phi=B_2$, each root has a root orthogonal to it, so for every root there is some nontrivial element (in fact, a reflection) of the Weyl group fixing it. But e.g. a simple root in $B_2$ is not fixed by any simple reflection. However, what you might want to know is the following. If we choose any point $...


4

No, it is not true. Take $\mathfrak{sl}_3$ with simple reflections $r, s$. Then the longest element $t := rsr=srs$ is also a reflection. Put $w=e$. Then $w=e < sw=s < stw = rs$, but $tw = srs \nless stw = rs$.


4

First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include the special unitary groups (etc.). which live over many fields. While Steinberg's lecture notes are often quite useful, a more leisurely treatment (often ...


4

The cyclicity of the Soergel modules (the ideals you write are principal) is equivalent to rational smoothness of the corresponding Schubert varieties. Or more generally, this can be written as the following condition on Kazhdan-Lusztig polynomials $P_{xy}(q)=q^{l(x)-l(y)}$, which works e.g. for dihedral groups and any $x,y$. See for example "4." ...


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