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2 results

  • Fundamental local equivalences in quantum geometric Langlands

  • Part of
  • Justin Campbell, Gurbir Dhillon, Sam Raskin
  • Journal:
    Compositio Mathematica / Volume 157 / Issue 12 / December 2021
    Published online by Cambridge University Press:
    27 December 2021, pp. 2699-2732
    Print publication:
    December 2021
    • Article
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      • Open access
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  • In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory and Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.

  • $D$-modules on spaces of rational maps

  • Part of
  • Jonathan Barlev
  • Journal:
    Compositio Mathematica / Volume 150 / Issue 5 / May 2014
    Published online by Cambridge University Press:
    31 March 2014, pp. 835-876
    Print publication:
    May 2014
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  • Let $X$ be an algebraic curve. We study the problem of parametrizing geometric structures over $X$ which are only generically defined. For example, parametrizing generically defined maps (rational maps) from $X$ to a fixed target scheme $Y$. There are three methods for constructing functors of points for such moduli problems (all originally due to Drinfeld), and we show that the resulting functors are equivalent in the fppf Grothendieck topology. As an application, we obtain three presentations for the category of $D$-modules ‘on’ $B(K)\backslash G(\mathbb{A})/G(\mathbb{O})$, and we combine results about this category coming from the different presentations.