Several questions on semi infinite flag manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:16:43Z http://mathoverflow.net/feeds/question/77107 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77107/several-questions-on-semi-infinite-flag-manifold Several questions on semi infinite flag manifold Shizhuo Zhang 2011-10-04T06:03:45Z 2011-10-05T00:10:19Z <p>Let $G$ be a connected simply-connected Lie group correspondence to simple Lie algebra $g$,consider the loop algebra $g((t))$ and so called "Natural Borel subalgebra" $n[t,t^{-1}]\oplus h[t]$,denoted by $\mathfrak{b}$ and consider ind-group $G((t))$ associated to $g((t))$ and ind group $N_{-}((t))$ and $H[[t]]$ correspondence to completion of $n_{-}[t,t^{-1}]$ and $h[t]$ in $g((t))$.</p> <p>Look at the "Natural Borel subgroup" $B$ correspondence to "natural borel subalgebra" $\mathfrak{b}$</p> <p>It is equal to $N_{-}((t))$$H[[t]]$. Now consider the quotient $X:= G((t))/B$. Frenkel and Ben zvi claimed that $X$ can not be given a scheme or an ind-scheme structures. </p> <p>My first question is How to see this? For example, if $G=SL_2(\mathbb{C})$, how to see $X$ can not be given a scheme structure? Can one use analogue of Birkhoff decomposition to give a scheme structure(the $w$-translate of big "semi-inifnite" cell form an open affine cover and algebra of regular functions on these big cell should identify with contragradient Wakimoto modules)</p> <p>Then they claimed that $X$ should be viewed as formal loop space of finite dimensional flag variety $G/B_{-}$, which is $Hom(Spec\mathbb{C}((t)),G/B_{-})$. I wonder whether this statement is equivalent to say they are "isomorphic" to each other(in what sense?) and how to prove this statement? </p> <p>Any hint and related comments is welcome. </p> <p>Thanks!</p> http://mathoverflow.net/questions/77107/several-questions-on-semi-infinite-flag-manifold/77168#77168 Answer by Reimundo Heluani for Several questions on semi infinite flag manifold Reimundo Heluani 2011-10-04T19:58:06Z 2011-10-05T00:10:19Z <p>Until you get a better answer this may help you. As far as I understand the semi-infinite flag manifold appeared in Feigin-Frenkel's paper [1] where they weren't really defined as an algebro-geometric object but rather they constructed what morally should be called some sheaves on them. Since both $G(K)$ and $N_-(K)\cdot H(\mathcal{O})$ are ind-groups, a priori their quotient as $k$-spaces is the sheafification of the quotient functor and it is not clear if it has a stratification by finite type schemes. I use the words <em>finite type</em> here because these spaces were introduced in the hopes that some category of perverse sheaves on them would be equivalent to some category of representations of the affine algebra for $g$ (generalizing Beilinson-Bernstein localization)(as Alexander Braverman noted below, we could deal with finite codimention strata as well). I quote from [2] (slightly different setup than yours)</p> <p><em>...Since the pioneering work of Feigin and Frenkel [1], people were trying to develop the theory of perverse sheaves (constructible with respect to a given stratification) on G((t))/B((t)) ... The problem is that G((t))/B((t)) is very essentially infinite-dimensional, so that the conventional theory of perverse sheaves, defined for schemes of finite type, was not applicable. Since it was (and still is) not clear whether there exists a direct definition of perverse sheaves on G((t))/B((t))...</em> </p> <p>And similar quote from [3]<br> <em>The semi-infinite flag manifold, thought of as $G(K )/N (K ) · T (\mathcal{O} )$, does not carry an algebro-geometric structure that would allow for the theory of perverse sheaves, or D-modules, in the way it is known today.</em></p> <p>So it is my understanding that there has been work in trying to find inductive systems of schemes without success, in the case of the affine Grassmanian the situation is simplified because the $G(\mathcal{O})$ orbits are finite dimensional and there is a lattice model for the $GL(n)$ case. The general case and the affine flag case (not semi-infinite, but quotient by Iwahori) can be reduced to this $GL(n)$ case. In the semi-infinite situation, all orbits are infinite dimensional. The above mentioned article [3] in fact does construct a category that has all the properties that we would like perverse sheaves on $X$ to have, to that end the authors use an actual ind-scheme that serves to approximate $X$ (they use $Bun N$). </p> <p>As for your second question regarding maps from a curve (the punctured disk) to the flag manifold, I'll refer you to section 3 and 4 chapter 1 (arxiv version) of [4] which by the way is the first attempt (as far as I know) of writing a category of perverse sheaves on these spaces. There's also a discussion there on the ind-scheme structures of related spaces constructed globally on a curve. </p> <p>[1] B. Feigin, E. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990) 161–189.<br> [2] Braverman, A. Finkelberg, M.; Gaitsgory, D. Mirković, I. Iersection cohomology of Drinfeld's compactifications. Selecta Math. (N.S.) 8 (2002), no. 3, 381–418.<br> [3] Arkhipov, S. Braverman, A. Bezrukavnikov, R. Gaitsgory, D. Mirković, I. Modules over the small quantum group and semi-infinite flag manifold. Transform. Groups 10 (2005), no. 3-4, 279–362.<br> [4] Finkelberg, Michael; Mirković, Ivan Semi-infinite flags. I. Case of global curve P1. Differential topology, infinite-dimensional Lie algebras, and applications, 81–112, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999. </p> http://mathoverflow.net/questions/77107/several-questions-on-semi-infinite-flag-manifold/77184#77184 Answer by Alexander Braverman for Several questions on semi infinite flag manifold Alexander Braverman 2011-10-04T23:18:39Z 2011-10-04T23:18:39Z <p>About defining the (ind)scheme structure: working with particular strata is basically never a good way to do this. What you need in order to define an algebro-geometric object is to define a functor from $Schemes$ to $Sets$ that it represents (it is enough to do it for affine schemes, i.e. it is enough to say what is an $R$-point of your space when $R$ is a ring). This is easy to do for semi-infinite flags. After you have done this, you can ask whether this functor is representable by a scheme or an ind-scheme (but I want to emphasize that this question doesn't make sense before you define the functor).</p>