Talk:Borel–Weil–Bott theorem

I seem to have failed with the TeX at the end. Charles Matthews 18:28, 14 Jun 2004 (UTC)

Propose Move to Borel-Weil-Bott theorem

I propose to move this page to Borel-Weil-Bott theorem. This is the more common name (google scholar returned 521 hits versus 37 for the "Borel-Bott-Weil theorem"), and also follows standard mathematical theorem naming convention in that it is an extension by Bott of a theorem of Borel and Weil. RobHar (talk) 05:02, 29 July 2008 (UTC)

Done, and fixed double redirects. RobHar (talk) 17:58, 10 January 2009 (UTC)

Serious Typo

There's a serious typo in this article, namely the definition of "dominant." A weight $\mu$ is dominant iff $(\mu ,\alpha )\geq 0$ for all simple roots $\alpha$; but in the article it states that a weight $\mu$ is dominant iff $(\mu ,\alpha )\gg 0$ for all simple $\alpha$, which is false. It makes sense how this typo occurred, though, since it IS true that for the theorem one wants to choose w such that $(w(\lambda +\rho ),\alpha )>0$. However, this is NOT the same as $w(\lambda +\rho )$ being dominant; it's the same as $w*\lambda$ being dominant, where $w*$ denotes the "twisted" Weyl group action centered at $\rho$. This is a common error when discussing this theorem; personally, I think it's better exposition to define the twisted action and then say "let w be the unique element of the Weyl group such that $w*\lambda$ is dominant," since this is almost always how this theorem is stated in practice. Also, one should avoid using $\gg$ at all here.

I think also something that's implicit in this article should be made explicit: This theorem is valid only for algebraic groups or Lie groups over the complex numbers. It is worth emphasizing this because in the theory of algebraic groups one also considers algebraic groups over fields of positive characteristic, and a strictly weaker form of this theorem holds there. In particular, this theorem is false in its full generality there, but in fact it is still a huge open problem in representation theory to determine exactly what does hold.

Since this is my first edit on Wikipedia, I've erred on the side of being cautious and not actually making any of these changes myself. Should I just go ahead and make them? FrobeniusTwist (talk) 16:02, 15 March 2009 (UTC)

It certainly sounds like you know what you're talking about (and what your saying is true), so I would indeed encourage you to make the appropriate changes. Welcome to wikipedia! RobHar (talk) 16:51, 15 March 2009 (UTC)

Okay, I made some changes; let me know if they look okay! FrobeniusTwist (talk) 01:49, 16 March 2009 (UTC)

They look pretty good. But there was a typo in the section on positive characteristic. If $\lambda$ is non-dominant and is not in the dotted Weyl orbit of a dominant weight, cohomology may not vanish. It does vanish if the weight is close to zero, as explained in Part II of Jantzen's book, Corollary 5.5.

Merge with Borel–Weil theorem

There is no rational reason to have separate articles. I will do it when I get enough time. Arcfrk (talk) 21:13, 12 April 2009 (UTC)

I'd suggest merge Borel–Weil theorem into this article, but maybe I'm just crazy. RobHar (talk) 15:50, 13 April 2009 (UTC)