Question

My questions here are from my attempt at trying to understand the definition on pg 15 in [FG2]-"Local Geometric Langlands Correspondence & Affine Kac-Moody Algebras" (http://arxiv.org/PS_cache/math/pdf/0508/0508382v3.pdf ). It is also discussed on pg 122 in http://math.berkeley.edu/~frenkel/loop.pdf (and in some other places such as Beilinson-Drinfeldt's "Quantization of Hitchin's integrable system & Hecke eigensheaves").

Question 1: In this context, what precisely is a connection $\nabla$ on a principal $G$-bundle $\mathfrak{F}_G$ on $X$? (Here $X$ is a smooth curve, or the formal disc, or the punctured disc.) Here, a principal $G$-bundle $\mathfrak{F}_G$ is presumably defined as a scheme with a map $\mathfrak{F}_G \rightarrow X$, a $G$-action $G \times \mathfrak{F}_G \rightarrow \mathfrak{F}_G$, satisfying certain compatibilities.

Question 2: In the same paragraph: given a reduction $\mathfrak{F}_B$ of $\mathfrak{F}_G$ to the Borel $B$, what does it mean for $\nabla$ to preserve $\mathfrak{F}_B$?

What does the notation $\mathfrak{g}_{F_G}$ mean, and why is $ \nabla - \nabla' \in \mathfrak{g}_{F_G}$? Finally, how to project $\nabla - \nabla'$ to $(\mathfrak{g}/\mathfrak{b})_{F_B} \times \omega_X$?

Question 3: In the paragraph on pg 15 after the Lemma, is the $B$-bundle $\mathfrak{F}_B$ a trivial bundle? (I am confused by this, because not all bundles on a smooth curve are trivial). After picking a trivialization, how do we get a tautological connection $\nabla^{0}$?

Question 4: On pg 16, Section 1.2, when $R$ is a $D_X$ algebra, how is an $R$-family of $\mathfrak{g}$-opers defined?

Answer 1

Question 1: In the answers to a previous question of yours, I think it was explained that one way to view a connection is as infinitesimal descent data. It might be more useful, however, to look at some specific examples: when $G=GL_n$, then a $G$-bundle with flat connection is just a vector bundle of rank $n$ with a flat connection; when $G=SL_n$, it's a vector bundle $V$ of rank $n$ equipped with a flat connection $\nabla$ such that $\det(V)$ admits a $\nabla$-parallel generating section; when $G=SO_n$, it's just a $SL_n$-bundle with flat connection $\nabla$ equipped also with a $\nabla$-parallel non-degenerate quadratic form....

Question 2: In any of the above explicit settings, giving a reduction to a Borel $B$ simply amounts to giving an appropriate full flag of sub-bundles for $V$ (for example, if $G=SO_n$, you will require some isotropicity condition for the flag). A connection will preserve the reduction $\mathfrak{F}_B$ if it preserves the flag it corresponds to (in the sense that every derivation preserves the flag). If $\nabla$ and $\nabla'$ are two flat connections (of $G$-bundles), then their difference is an element of $End(V)\otimes\Omega^1$. Of course, their difference will kill any structures on $V$ that are parallel with respect to both $\nabla$ and $\nabla'$ : so, if $G=SL_n$, then it will be of trace $0$; if $G=SO_n$, then it will be of trace $0$ and preserve the quadratic form. This sub-space, which is a twist of $\mathfrak{g}=Lie(G)$ (viewed as a representation of $G$) by the $G$-torsor $\mathfrak{F}$, is what is being referred to as $\mathfrak{g}_{\mathfrak{F}_G}$.

So we find that $\nabla-\nabla'$ is an element of $\mathfrak{g}_{\mathfrak{F}_G}\otimes\Omega^1$.

EDIT: I was trying to give an idea of what preserving $\mathfrak{F}_B$ meant, but the notion of $\nabla$ preserving $\mathfrak{F}_B$ is simple in general: we have an identification of $G$-torsors: $$\mathfrak{F}_G=G\times^B\mathfrak{F}_B.$$ We now simply require the connection $\nabla$ to arise from a connection on $\mathfrak{F}_B$ via this identification.

Question 3: Here, I think they're only asking for a local trivialization of $\mathfrak{F}_B$. A trivialization of $\mathfrak{F}_B$ gives an isomorphism of $B$-torsors from $B\times X$ to $\mathfrak{F}_B$. The former has a tautological connection over $X$, since it's a constant $X$-scheme. EDIT: The local trivialization gives a local tautological connection!

Question 4: No idea.

EDIT: In general, if you have a $G$-torsor $P$ over $X$, and a representation $V$ of $G$, you can define the twist $V_P$ as a vector bundle over $X$: this is the contraction product $$V\times^G P=(V\times P)/G,$$ where $G$ is acting diagonally. If you look at 1.2.4 in Frenkel's book ('Langlands correspondence...'), then you'll find that giving a connection on $P$ is equivalent to giving compatible flat connections on the twists $V_P$ of all algebraic $G$-representations $V$. Essentially, if you have infinitesimal descent data for $P$, then you clearly have corresponding data for the twists $V_P$; the converse is trickier and uses some Tannakian theory. When $G$ has a faithful representation $V$ (like in our examples), it is enough to do this for the one twist $V_P$ (again, by the Tannakian machine).

Answer 2

This is just a small remark about how to make sense of opers parametrized by a $\mathcal{D}_X$-algebra $R$. [And admittedly I'm not actually sure it's correct, but it seems to be the "obvious" way to do it.]

Let's just assume for the moment that $G=GL(n)$, since as in Keerthi's answer by Tannakian nonsense this suffices. Then $\mathcal{D}_X\otimes_{\mathcal{O}_X}R$ has a natural structure of a (noncommutative) sheaf of algebras, where one uses the $\mathcal{D}_X$-action on $R$ to define commutators between elements of $R$ and $\mathcal{D}_X$; and it comes equipped with a (noncentral) algebra homomorphism $R\rightarrow \mathcal{D}_X\otimes_{\mathcal{O}_X}R$ (analogous to the usual $\mathcal{O}_X\rightarrow \mathcal{D}_X$). A connection on a family $V$ of vector bundles parametrized by the $\mathcal{D}_X$-algebra $R$ (in other words, $V$ is a locally projective $R$-module) should be an extension of the $R$-action on $V$ to an $\mathcal{D}_X\otimes_{\mathcal{O}_X}R$-action on $V$. Compatibility of a filtration with the connection is just as Keerthi describes it.

This will be equivalent to the crystalline description: giving $R$ a $\mathcal{D}_X$-algebra structure is equivalent to descending it to a sheaf of algebras on the de Rham space of $X$ (i.e. descending $\operatorname{Spec}(R)$ to a space $\mathcal{R}$ affine over the de Rham space of $X$), and giving $V$ an action of $\mathcal{D}_X\otimes_{\mathcal{O}_X}R$ will be equivalent to giving descent data to a vector bundle on $\mathcal{R}$.