Timeline for Rep Theory Consequences of Bott--Weil--Borel
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2013 at 15:44 | answer | added | Chuck Hague | timeline score: 6 | |
| Feb 7, 2013 at 13:39 | answer | added | Allen Knutson | timeline score: 24 | |
| Feb 7, 2013 at 10:28 | vote | accept | Jean Delinez | ||
| Feb 7, 2013 at 2:06 | comment | added | Jan Weidner | I think one can regard the BWB theorem as a precursor of the Beilinson Bernstein localization theorem. The latter gives an equivalence between representations of the Lie algebra and D-modules on the flag variety. I think one can hardly overestimate the importance of the latter theorem. It allows to use a lot of geometric machinery to solve representation theoretic problems. For example it was a key ingredient in the original proof of the Kazhdan-Lusztig conjectures. | |
| Feb 6, 2013 at 18:23 | answer | added | Jim Humphreys | timeline score: 39 | |
| Feb 6, 2013 at 18:19 | comment | added | Marty | First, I don't think that representations are coming from nowhere. When a group $G$ acts on a space $X$, and you have a $G$-equivariant bundle on the space $X$, then you get a representation of $G$ on the sections (and higher cohomology) of the bundle. Maybe the most impressive results beyond the theorem itself are how useful it is for generalizations. The generalization that comes first to my mind is Schmid's "L²-cohomology and the discrete series" (Annals, 1976) which proved a conjecture of Langlands by using a geometric realization in the spirit of Borel-Weil-Bott. | |
| Feb 6, 2013 at 16:35 | comment | added | Venkataramana | You can take a look at Shrawan Kumar's paper on the PRV conjecture, which was a conjecture about how to decompose the tensor product of two irreducible finite dimensional representations of a complex simple group$G$. The proof heavily uses the geometry of $G/B\times G/B$ the flag variety of $G\times G$. (Shravan Kumar, Invent math ,1988, vol 93, 117-130. There is also the standard monomial theory, a collection of results in representation theory, proved using geometric methods. | |
| Feb 6, 2013 at 16:18 | history | asked | Jean Delinez | CC BY-SA 3.0 |