Chiral principal series categories. I: Finite dimensional calculations
This manuscript begins a series studying \(D\)-modules on the Feigin-Frenkel semi-infinite flag variety from the perspective of the Beilinson-Drinfeld factorization, or chiral, theory. Let \(G\) be a split reductive group over a field \(k\) whose \(\text{char}(k)=0\). Let \(X\) be a smooth curve and let \(x\in X\) be a fixed \(k\)-point. Let \(O_x=kt_x\) be the ring of Taylor series based at \(x\), and \(K_x = k((t_x))\) be the ring of Laurent series based at \(x\). Zastava spaces are certain finite-dimensional subvarieties of the affine Grassmannian (Section 2). S. Raskin calculates Whittaker-twisted cohomology groups of Zastava spaces in Theorem 4.6.1. The author shows that such cohomology groups realize the nilradical of a Borel subalgebra for the Langlands dual group, following earlier work of \textit{B. Feigin} et al. [Transl., Ser. 2, Am. Math. Soc. 194, 113--148 (1999; Zbl 1076.14511)] and \textit{A. Braverman} and \textit{D. Gaitsgory} [Invent. Math. 150, No. 2, 287--384 (2002; Zbl 1046.11048); Geom. Funct. Anal. 17, No. 6, 1788--1850 (2008; Zbl 1234.11155)]. He compares this geometric realization of the Langlands dual group to the standard one provided by factorizable geometric Satake in Theorem 7.9.1.
scientific article; zbMATH DE number 7374911
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