Timeline for Weight multiplicity formulae for $(\mathfrak g,B)$-irreps
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 10, 2014 at 18:58 | answer | added | Jim Humphreys | timeline score: 3 | |
| Oct 6, 2014 at 21:17 | comment | added | Allen Knutson | Exists a formula, or should exist such a formula? Do you think a reference exists? (This is really what motivates my original question.) | |
| Oct 6, 2014 at 20:10 | comment | added | Victor Protsak | The expression of the character via the KL conjecture is kind of orthogonal to the question: for one thing, in the dominant integral case it does not yield a positive formula; for another, for special Weyl chambers (e.g. if w=w_P as in my answer below), there is a straightforward positive expansion of the character without any resort to KL theory. In fact, I believe that in the gl case there exists a combinatorial formula for every chamber based on GZ patterns. | |
| Oct 5, 2014 at 0:12 | comment | added | Allen Knutson | (1) When $G\geq H$, a $(\mathfrak{g},H)$-module $V$ is one with actions of $\mathfrak{g}$ and of $H$, from each of which one can derive an action of $\mathfrak h$; the two actions should agree. (2) As for what extra information it should contain beyond that of a $\mathfrak{g}$-representation: a $(\mathfrak g,B)$-irrep is in category $\mathcal O$, unless I'm very confused. (3) My weight $\lambda$ was assumed integral, so better to think of it as a weight for $T$ (or $B$ since $B/[B,B] \equiv T$) than for $\mathfrak t$. | |
| Oct 4, 2014 at 20:34 | comment | added | Jim Humphreys | P.S. I still don't understand how the notion of $(\mathfrak{g}, B)$-module is defined, or what extra information it might contain. Note that a "weight" for $\mathfrak{g}$ (or a Cartan subalgebra) is the differential of a weight for a torus. | |
| Oct 4, 2014 at 2:56 | comment | added | Alexander Braverman | I second what Jim says - I doubt that there is anything better than what is given by the Kazhdan-Lusztig conjecture. | |
| Oct 4, 2014 at 0:53 | history | edited | Allen Knutson | CC BY-SA 3.0 |
added 10 characters in body
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| Oct 4, 2014 at 0:53 | comment | added | Allen Knutson | I meant to have reductive there; I'll put it in. Yes your $L(\lambda)$ is my $V_\lambda$, insofar as they're unique. What could there be to say about "the nature of the $B$-action"? 1-d reps are of $B/B' \cong T$, so weights, like $\lambda$. | |
| Oct 3, 2014 at 22:21 | answer | added | Victor Protsak | timeline score: 2 | |
| Oct 3, 2014 at 20:46 | comment | added | Jim Humphreys | Can you explain a bit more about the nature of the $B$-action? In any case, it seems that your $V_\lambda$ is the usual irreducible $\mathfrak{g}$-quotient of the Verma module (often denoted $L(\lambda)$). The Kazhdan-Lusztig algorithm gives a sort of "combinatorial formula", far from positive, for the weight multiplicities; but it's extremely hard to compute in most cases. So I'm not optimistic. (By the way, I guess $G$ here should be semisimple?) | |
| Oct 3, 2014 at 20:07 | history | asked | Allen Knutson | CC BY-SA 3.0 |