Decomposition of the ring of functions on the unipotent radical of a Borel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:38:11Z http://mathoverflow.net/feeds/question/88567 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88567/decomposition-of-the-ring-of-functions-on-the-unipotent-radical-of-a-borel Decomposition of the ring of functions on the unipotent radical of a Borel Chuck Hague 2012-02-15T21:53:27Z 2012-02-16T13:26:57Z <p><strong>Background</strong></p> <p>Let $G$ be a semisimple linear algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B \subseteq G$ be a Borel subgroup of $G$ and let $U \subseteq B$ be its unipotent radical. Consider $k[U]$ as a $U$-module under left multiplication in $U$; let's call this module $k[U]_L$. By, say, identifying $U$ with the big cell in the flag variety $G/B^-$, it is not hard to see that $k[U]_L$ is isomorphic as a $U$-module to a direct limit of standard modules $H^0(\lambda)$ for $G$. (In characteristic 0 one can also see this by identifying $k[U]$ with the dual zero Verma module for the enveloping algebra of $G$, cf <a href="http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00347-7/home.html" rel="nofollow">this paper</a>). In fact there is even a natural ring structure on this direct limit such that this isomorphism is a ring isomorphism.</p> <p>EDIT: A reference for this fact is Lemma 2.5 and the discussion following it in "<a href="http://www.kryakin.com/files/Invent_mat_%282_8%29/123/123_06.pdf" rel="nofollow">The Nil Hecke Ring and Singularity of Schubert Varieties</a>," by Shrawan Kumar. Although the construction is done there over $\mathbb C$, it works over any algebraically closed field, and although Kumar only proves that there is a $T$-equivariant morphism, it is easy to check that the morphism he constructs is $U$-equivariant.</p> <p><strong>Question</strong></p> <p>Now consider $k[U]$ as a $U$-module under the <em>conjugation</em> action of $U$ on itself; let's call this module $k[U]_C$. Is there a nice description of $k[U]_C$ as a $U$-module in terms of $G$-modules in a way analogous to the description of $k[U]_L$?</p> http://mathoverflow.net/questions/88567/decomposition-of-the-ring-of-functions-on-the-unipotent-radical-of-a-borel/88572#88572 Answer by Alexander Braverman for Decomposition of the ring of functions on the unipotent radical of a Borel Alexander Braverman 2012-02-15T22:42:00Z 2012-02-15T22:42:00Z <p>It is probably hard to give a complete description. A lot of partial information about this is contained in this paper of Kostant <a href="http://front.math.ucdavis.edu/1201.4494" rel="nofollow">http://front.math.ucdavis.edu/1201.4494</a> (in particular there is a description of the subring of invariant elements in $k[U]$; in fact Kostant works with the Lie algebra of $U$ instead of $U$ itself, but clearly they are isomorphic as $U$-varieties with respect to the adjoint action).</p> http://mathoverflow.net/questions/88567/decomposition-of-the-ring-of-functions-on-the-unipotent-radical-of-a-borel/88575#88575 Answer by Jim Humphreys for Decomposition of the ring of functions on the unipotent radical of a Borel Jim Humphreys 2012-02-15T23:12:13Z 2012-02-16T13:26:57Z <p>At first sight it's unclear how much the question has to do with the ambient semisimple group <code>$G$</code>: consider the simplest case when the rank is 1 and <code>$U$</code> is just the additive group. It's true that the conjugation action of <code>$G$</code> itself on its function algebra has a rich structure. This was shown in characteristic 0 by Kostant and then in a more algebraic setting by Richardson, after which Steve Donkin (Invent. Math. 91, 1988) generalized their results to almost all prime characteristics; his paper is freely available online at</p> <p><a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002104814" rel="nofollow">http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002104814</a></p> <p>The maximal unipotent subgroup <code>$U$</code> plays an essential role in the study of a semisimple group, but as a variety it just has the structure of an affine space. So its actions on its own functions won't provide direct information related to the representations of <code>$G$</code>.</p> <p>ADDED: The answer to the question as stated is no, based on the rank 1 case where there is no useful connection between the (trivial) conjugation action of <code>$U$</code> and the representation theory of <code>$G$</code>. Even with the added references, I haven't been able to figure out what is really being asked in this smallest case. It would help to start with a more explicit formulation when <code>$G = SL_2$</code>.</p>