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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.

MSC:
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
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