32,427 reputation
369144
bio website math.umass.edu/~jeh
location U. Massachusetts, Amherst
age 75
visits member for 5 years, 4 months
seen 3 mins ago
More-or-less retired professor at UMass Amherst. Basically an algebraist with interests in Lie theory, specifically representation theory and related algebraic geometry.

1d
comment What is Jantzen's formula for the determinant of the Shapovalov form in the case of generalized Verma modules?
P.S. The basic paper by Jantzen investigating the parabolic case is available here: gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002313782 (see especially Satz 2, where there is a lot of notation which I'll try to unpack in your setting). Keep in mind that the determinant on each weight space is only computed up to an arbitrary nonzero constant in $\mathbb{C}$.
1d
comment What is Jantzen's formula for the determinant of the Shapovalov form in the case of generalized Verma modules?
Two preliminary comments: 1) Kac-Kazhdan work in a vastly more general setting, so it's probably not useful to consult their concise paper if your question is about a finite dimensional case. 2) I'm unclear about your notation and "$d$-dimensional": $SO(d)$ usually means a compact Lie group of type $B_\ell$ or $D_\ell$ for $d=2\ell+1$ or $2\ell$, so you'd have to pass to the corresponding complex Lie algebra. In that case, what is the smallest $d$ you'd be interested in? (I'm trying to pin down the parabolic subalgebras and Levi subalgebras involved.)
1d
comment Cohomology of Flag Varieties
I'm not sure precisely what should be attributed to Leray, though it should be noted that in his 1952 thesis (with Leray on the examining committee) Borel does discuss the foundational contributions of Leray and others while emphasizing that his own viewpoint is different in crucial ways. It might be useful to sort out the history more carefully than I have, but Borel's formulation seems to have had more widespread influence in later work on compact Lie groups and homogeneous spaces.
Jun
29
revised Is every weight of an integrable highest weight module in the Tits cone?
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Jun
29
comment Is every weight of an integrable highest weight module in the Tits cone?
What do you mean by "the" Weyl chamber (or by the Tits cone)? Aside from that, it's a good idea to start with familiar examples of integrable modules such as the adjoint representation: its weights are the roots (not necessarily all real) together with 0.
Jun
24
comment Why are parabolic subgroups called “parabolic subgroups”?
@LSpice: Mais oui!
Jun
23
comment Can the product of a simple and a non-simple indecomposable representation be semisimple?
I'm not sure what the answer is, but does it have consequences (either way)? There tend to be lots of exotic infinite dimensional representations of such Lie algebras, so it's hard to predict what is possible. There don't seem to be examples in the BGG category of such semisimple tensor products with $\rho$ finite dimensional and $\pi$ of course not; but this category of modules is very restrictive. If I had to guess, I'd expect to find no examples of the type you describe.
Jun
23
revised On the vertices of a Coxeter complex
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Jun
23
revised On the vertices of a Coxeter complex
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Jun
23
answered On the vertices of a Coxeter complex
Jun
22
comment Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?
@slider: It may (or may not) help to look at my short notes on Weyl modules posted at my homepage: people.math.umass.edu/~jeh/pub/weyl.pdf. By now the notation and terminology have shifted a lot, so the history gets a bit lost. But the notion of "tilting module" adds to the complexity. I should emphasize that higher sheaf cohomology modules attached to line bundles are still quite hard to study, though I've proposed some approaches using inverse Kazhdan-Lusztig polynomials for affine Weyl groups.
Jun
22
answered Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?
Jun
22
comment Details on the Symmetric Group action on chambers of the Shi Arrangement
By the way, the published version of this paper is freely available online: sciencedirect.com/science/article/pii/S0012365X98003653
Jun
18
comment The relation of nilpotent orbits and simple singularities, for orbits smaller than subregular ones
Here's the direct link to the preprint Daniel discusses: front.math.ucdavis.edu/1502.05770
Jun
17
comment How to embed $U(1)$ into a bigger group, using Dynkin diagrams
Yes, this is where it all gets confusing for me. I'm still uncertain where the embedding comes from in your second paragraph.
Jun
17
awarded  Good Answer
Jun
17
comment How to embed $U(1)$ into a bigger group, using Dynkin diagrams
Concerning the proposed embedding in your second paragraph, why do you think this should exist?
Jun
17
comment How to embed $U(1)$ into a bigger group, using Dynkin diagrams
P.S. Maybe I should add that the pseudo-Levi subgroups of maximal rank (say in a compact Lie group) provide the needed centralizers of semisimple elements (which applies to all elements of a compact Lie group). Strictly speaking, in some situations these centralizers may fail to be connected, but for example not in the simply connected groups.
Jun
17
comment How to embed $U(1)$ into a bigger group, using Dynkin diagrams
What you are looking for originates in the older work of Borel and deSiebenthal: gdz.sub.uni-goettingen.de/dms/load/img/… (which figures in some previous questions here that you can find by searching for "Siebenthal"). The main point is to look for subdiagrams of the extended Dynkin diagram, as Andre indicates in a relevant example. Here you get "pseudo-Levi subgroups", which are not necessarily actual Levi factors in parabolic subgroups of the given group.
Jun
15
revised Evaluation of Macdonald Polynomials at $x_1=x_2=…=x_n = h$
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