Higher K theory and algebraic cycles in representation theory? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:37:40Z http://mathoverflow.net/feeds/question/57972 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57972/higher-k-theory-and-algebraic-cycles-in-representation-theory Higher K theory and algebraic cycles in representation theory? Shizhuo Zhang 2011-03-09T17:21:07Z 2011-03-10T15:12:50Z <p>Can anybody talk about how Higher K theory and algebraic cycles play roles in representation theory? I am more interested in how they play roles in Kazhdan-Lusztig conjectures.</p> <p>Of course K_0 plays important roles in representation theory,but I don't know whether Higher K theory is also useful in representation theory.</p> <p>Any references will be very welcome!</p> <p>Thanks in advance</p> http://mathoverflow.net/questions/57972/higher-k-theory-and-algebraic-cycles-in-representation-theory/58032#58032 Answer by David Ben-Zvi for Higher K theory and algebraic cycles in representation theory? David Ben-Zvi 2011-03-10T02:55:36Z 2011-03-10T15:12:50Z <p>The \$K_2\$ functor has an important role in representation theory, starting from works of Bloch in which he identified the Kac-Moody central extension of loop algebras in terms of \$K_2\$ - a more updated form of this is the IHES paper of Deligne-Brylinski about \$K_2\$ central extensions of reductive groups. The universal class in \$H^4(BG,Z)\$ for a simple group \$G\$ which is responsible for the level in Kac-Moody algebras/Chern-Simons theory or the \$q\$ in quantum groups etc can be interpreted as representing this \$K_2\$ central extension. Anyway it's a very beautiful story we haven't really fully seen the impact of yet (but is also behind eg the famous constructions of Fock-Goncharov in Teichmuller theory).</p> <p>Anyway that's not the kind of role you're talking about, in which we take higher K groups of a category of representations. One way to answer that is to say certainly the full homotopy theory of a category of representations is very important - but there we usually ask for something more subtle, ie identifying the entire category itself in some geometric way rather than just its \$K\$-theory spectrum or the homotopy groups of the latter. To really see higher K-groups arising the way you're asking I think you'd need to consider problems in which FAMILIES of representations play a central role -- single representations (or contractible families thereof) are measured by \$K_0\$, but if you are interested in measuring families over a circle (ie automorphisms) or over more complicated bases (eg higher spheres - i.e. \$n\$-automorphisms), these would be measured by invariants in \$K_1\$ or higher \$K\$-groups.</p> <p>[Edit: When I said "something more subtle" I meant the following: n equivalence of (enhanced) derived categories gives rise to an isomorphism of K-theories -- eg Beilinson-Bernstein localization identifies K-groups of categories of Lie algebra representations with those of flag varieties. This certainly doesn't mean in general we know explicitly the K-groups! eg there are few categories we understand as well as \$Z\$-modules, but the corresponding K-groups are complicated! However it seems that these subtleties are arithmetic, and I don't see a clear representation theoretic role for them, though would be happy to learn otherwise.] </p> <p>The two other things that come to mind from your question are</p> <ul> <li><p>in Jacob Lurie's survey article on elliptic cohomology (available <a href="http://www.math.harvard.edu/~lurie/" rel="nofollow">here</a>) there's a theorem describing the K-theory spectrum of categories of integrable loop group representations in terms of a DAG version of nonabelian theta functions (in the spirit of results of Kac-Peterson, Looijenga and Ando)</p></li> <li><p>Beilinson's work on epsilon factors explains algebraic K-theory beautifully and applies it in a context related to automorphic forms.</p></li> </ul>