By Beilinson A., Drinfeld V.
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Additional resources for Quantization of Hitchin's integrable system and Hecke eigensheaves
Example text
Set N = [B, B], so H = B/N is the Cartan group. Denote by n ⊂ b ⊂ g, h = b/n the corresponding Lie algebras. g carries a canonical decreasing Lie algebra filtration gk such that g0 = b, g1 = n, and for any k > 0 the weights of the action of h = gr0 g on grk g (resp. gr−k g) are sums of k simple positive (resp. negative) roots. In particular gr−1 g = ⊕gα , α is a simple negative root. Set Z = ZG = CenterG. 2. Let X be any smooth (not necessarily complete) curve, FB a B- bundle on X. Denote by FG the induced G-torsor, so FB ⊂ FG .
0 denote the preimage of Bun0 Let LSG Gad in LSG . 1, so dim(T ∗ BunGad \T ∗ Bun0Gad ) < 58 A. BEILINSON AND V. DRINFELD dim T ∗ BunGad . The argument used in the proof of (42) shows that 0 ) < (2g − 2) · dim G + l. 2 one sees that LS 0 dim(LSG \ LSG G 0 of every is dense in LSG . So it suffices to prove that the preimage in LSG connected component of Bun0G is non-empty and irreducible. This is clear 0 → Bun0 is a torsor15 over T ∗ Bun0 . 12. On the Beauville – Laszlo Theorem. This section is, in fact, an appendix in which we explain a globalized version of the main theorem of [BLa95].
On the other hand it follows from the above lemma that if η has a zero then the dimension of YC at (F, η) is less than BunG . 4. Theorem. Nilp is Lagrangian. In this theorem we do not assume that g > 1. Proof. 2 we only have to show that Nilp has pure dimension dim BunG for g ≤ 1. 1) Let g = 0. Then Nilp = T ∗ BunG . A quasicompact open substack of BunG can be represented as H\M where M is a smooth variety and H is an HITCHIN’S INTEGRABLE SYSTEM 53 algebraic group acting on M . Then T ∗ (H\M ) = H\N where N ⊂ T ∗ M is the union of the conormal bundles of the orbits of H.