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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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