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Truncations of level 1 of elements in the loop group of a reductive group. (English) Zbl 1350.20035
Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(L\) be either \(k((t))\) or \(\mathrm{Quot}(W(k))\), the quotient field of the ring of Witt vectors of \(k\). Let \(\mathcal O\) be the valuation ring of \(L\). Similarly for a fixed \(q=p^r\) let \(F\) be either \(\mathbb F_q((t))\) or \(\mathrm{Quot}(W(\mathbb F_q))\), and \(\mathcal O_F\) be the valuation ring of \(F\). Let \(G\) be a connected reductive group over \(\mathcal O_F\), \(K\) be the subgroup \(G(\mathcal O)\) of \(G(L)\) and \(K_1\) be the kernel of the projection \(K\to G(k)\). Let \(\sigma\) be the Frobenius of \(L\) over \(F\). The \(\sigma\)-conjugacy class of \(b\in G(L)\) is the set \(\{g^{-1}b\sigma(g)\mid g\in G(L)\}\), and the \(K\)-\(\sigma\)-conjugacy class of \(b\) is \(\{g^{-1}b\sigma(b)\mid g\in K\}\).
In [R. E. Kottwitz, Compos. Math. 56, 201-220 (1985; Zbl 0597.20038)], the \(\sigma\)-conjugacy classes were classified by two invariants called the Newton point and the Kottwitz point, and in particular the set of \(K\)-\(\sigma\)-conjugacy classes was classified by a discrete invariant.
The paper under review studies a second invariant of \(K\)-\(\sigma\)-conjugacy classes, namely the truncation of level 1, which is defined as the associated \(K\)-\(\sigma\)-conjugacy class in \(K_1\backslash G(L)/K_1\). The author classifies truncations of level 1 in terms of Weyl group elements and dominant cocharacters of a maximal torus \(T\) in \(G\). The second main result of the paper compares the stratification of the loop group by truncations of level 1 to the stratification by \(\sigma\)-conjugacy classes. Finally as applications the author relates this invariant to the Ekedahl-Oort stratification of the Siegel moduli space and generalizations to other PEL Shimura varieties.

20G40 Linear algebraic groups over finite fields
20E45 Conjugacy classes for groups
14G35 Modular and Shimura varieties
22E67 Loop groups and related constructions, group-theoretic treatment
13F35 Witt vectors and related rings
Full Text: DOI arXiv
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