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asked Vanishing of power of nilpotent operator $\mathrm{ad} \, \;e$ in different characteritics
Apr
29
awarded  Nice Answer
Apr
18
comment Bruhat order of reflection subgroups
It would be worth asking Matthew Dyer at Notre Dame, since he is still active in this area. Besides proving (as did Deodhar) that a reflection subgroup of a Coxeter group is again a Coxeter group, he looked in an old paper at the Bruhat order as well: ams.org/mathscinet-getitem?mr=1104786 (This paper has been frequently cited, so probably some of its ideas have been followed up further; but I haven't kept track of all the research.)
Apr
17
revised structure of maximal tori in semisimple algebraic groups
deleted 185 characters in body
Apr
17
answered structure of maximal tori in semisimple algebraic groups
Apr
15
comment Do I understand the Chevalley Restriction Theorem correctly?
The answer to the question in the header (or the question in the text) is clearly no. I'm not sure the question is at research-level, but anyway it needs to be placed in the context of representations and Harish-Chandra's classical ideas: why does the restriction theorem matter? Working through the rank one case helps to see that the inverse process is subtle due in part to the fact that $\mathfrak{h}$ is abelian but $\mathfrak{g}$ is not. (See for instance $\S23$ of my 1972 textbook.)
Apr
1
comment Existence of lattices in reductive Lie groups
A comment on terminology here is that "reductive Lie group" doesn't seem like useful terminology, since a Lie group with an abelian Lie algebra might not be at all what most people think of as "reductive" in the Borel-Tits sense for algebraic groups. Borel and others did attempt to pin down what might be meant by a "real reductive Lie group", but not just in terms of the Lie algebra.
Apr
1
comment Cartan subspaces for general algebraic representations
The question you've formulated does look open-ended though worthwhile. Maybe it's simplest just to add this "answer" to the original question, as a remark?
Mar
30
answered Signs in Chevalley's commutator formula
Mar
30
revised Kazhdan-Lusztig graph for the Springer fiber of the minimal special unipotent class?
added 55 characters in body; edited title
Mar
27
comment Determining the Lie algebra elements exponentiating to the center of a Lie group
@Yannick: Thanks for the reference, though it doesn't help directly with the concrete question raised here. Hochschild studied carefully the general foundations for analytic groups and Lie groups, but without getting into the detailed study of special classes such as semisimple Lie groups (which probably need case-by-study at least initially)..
Mar
23
comment Is the Luna slice theorem valid for any orbit with a reductive stabilizer?
It would help if you made more explicit what you mean here by "affine space", as well as what kind of field you work over and what kind of source you are following for Luna slices. Aside from that, in the classical characteristic 0 setting for this theory, note that "reductive" is the same as "linearly reductive" in order to compare the passage Ariyan points to.
Mar
15
comment inductive construction of unipotent radicals
Also, your notion of "Coxeter diagram" is much more limited than the usual definition of "Coxeter graph" (or "Coxeter matrix") and creates ambiguity in passing to Lie algebras or groups: non-isomorphic ones may have isomorphic Coxeter groups attached.
Mar
10
comment Explicit formulas for certain elements in $Z(U(\mathfrak{gl_n}))$
This kind of question has come up earlier, so it's worth looking at some of the related ones, e.g., mathoverflow.net/questions/197786/…
Mar
10
answered indecomposable modules restricted from $gl_n$ to $sl_n$
Mar
10
comment inductive construction of unipotent radicals
The tag 'lie-groups' here seems inappropriate, since Lie groups don't in general have an intrinsic Jordan decomposition (in particular, "unipotent radical" isn't generally defined). Maybe 'kac-moody-algebras' or 'algebraic-groups'? So far there isn't a tag for Kac-Moody groups.
Mar
7
answered Characterization of restricted weights of representations of real semisimple Lie groups
Mar
6
comment reduction mod $p$ of Weyl modules
Can you clarify your set-up (and maybe motivation) a little more? For example, the concrete study of linear algebraic groups apparently intended here involves fields (not rings) of definition. Also, when you have the group $N_0$ acting on a module in prime characteristic, I guess you mean the reduction mod $p$ of this group? In your last paragraph, it's usual to view the $k$th symmetric power as the space of homogeneous polynomials in two variables of degree $k$, so your formulation looks nonstandard. (And you might add a tag 'algebraic-groups'.)
Mar
6
comment Can we count the number of simple modules for a reduced enveloping algebra?
I added a tag, partly to emphasize that the Lie algebras here come from algebraic groups and are studied using some of the geometry of the group. (By the way, the case of a trivial central character is easiest to study directly, but it leads to other central characters as well.)
Mar
6
revised Can we count the number of simple modules for a reduced enveloping algebra?
edited tags