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**Parametrizing nilpotent orbits via Bruhat-Tits theory.**
*(English)*
Zbl 1015.20033

Let \(k\) denote a field complete with respect to a nontrivial discrete valuation and having perfect residue field \(f\), let \(G\) denote the group of rational points of a reductive \(k\)-group, and let \(\mathfrak g\) denote the Lie algebra of \(G\). This paper parametrizes the nilpotent adjoint orbits of \(\mathfrak g\) in terms of data arising from the Bruhat-Tits building \(\mathcal B\) of \(G\).

To each point \(x\in \mathcal B\), A. Moy and G. Prasad [Invent. Math. 116, No. 1-3, 393-408 (1994; Zbl 0804.22008)], associate filtrations \(\{{\mathfrak g}_{x,r}\}\) and \(\{G_{x,r}\}\) of \(\mathfrak g\) and \(G\), respectively. Let \({\mathfrak g}_{x,r+}=\bigcup_{s>r}{\mathfrak g}_{x,s}\). The elements of \({\mathfrak g}_{x,r}/{\mathfrak g}_{x,r+}\) are called Moy-Prasad cosets of depth \(r\). Call such a coset “degenerate” if it contains a nilpotent element.

When \(r=0\), \({\mathfrak g}_{x,r}/{\mathfrak g}_{x,r+}\) is the Lie algebra \(L_x\) of the group of \(f\)-points of a certain connected reductive \(f\)-group. Thus, it makes sense to call a degenerate coset \(e\in{\mathfrak g}_{x,r}/{\mathfrak g}_{x,r+}\) “distinguished” if \(e\) is distinguished as an element of \(L_x\), i.e., \(e\) does not belong to any proper Levi subalgebra.

It turns out that for any \(r\in\mathbb{R}\), one can define what it means for a degenerate coset \(e\in{\mathfrak g}_{x,r}/{\mathfrak g}_{x,r+}\) to be distinguished. (Doing so requires partitioning the building \(\mathcal B\) into “generalized \(r\)-facets”. When \(r\) is an integer, these are just facets in the usual sense.) Let \(I_r^d\) denote the set of all distinguished, degenerate Moy-Prasad cosets of depth \(r\). The main theorem of this paper is that, modulo a certain equivalence relation, \(I_r^d\) parametrizes the set of nilpotent orbits in \(\mathfrak g\). Note that for each \(r\in\mathbb{R}\), one thus obtains a parametrization.

The result requires only mild hypotheses on \(k\) and \(G\). These assure the validity of, for example, a graded version of the Jacobson-Morozov Theorem. In particular, no explicit realization of the group \(G\) is required.

A few refinements to the parametrization are contained in joint work of the author and the reviewer [A generalization of a result of Kazhdan and Lusztig, Proc. Am. Math. Soc. (to appear)].

The motivating application of this paper is the author’s proof [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, No. 3, 391-422 (2002; Zbl 0999.22013)] of several homogeneity results for distributions on \(G\) and \(\mathfrak g\), including the Hales-Moy-Prasad Conjecture.

To each point \(x\in \mathcal B\), A. Moy and G. Prasad [Invent. Math. 116, No. 1-3, 393-408 (1994; Zbl 0804.22008)], associate filtrations \(\{{\mathfrak g}_{x,r}\}\) and \(\{G_{x,r}\}\) of \(\mathfrak g\) and \(G\), respectively. Let \({\mathfrak g}_{x,r+}=\bigcup_{s>r}{\mathfrak g}_{x,s}\). The elements of \({\mathfrak g}_{x,r}/{\mathfrak g}_{x,r+}\) are called Moy-Prasad cosets of depth \(r\). Call such a coset “degenerate” if it contains a nilpotent element.

When \(r=0\), \({\mathfrak g}_{x,r}/{\mathfrak g}_{x,r+}\) is the Lie algebra \(L_x\) of the group of \(f\)-points of a certain connected reductive \(f\)-group. Thus, it makes sense to call a degenerate coset \(e\in{\mathfrak g}_{x,r}/{\mathfrak g}_{x,r+}\) “distinguished” if \(e\) is distinguished as an element of \(L_x\), i.e., \(e\) does not belong to any proper Levi subalgebra.

It turns out that for any \(r\in\mathbb{R}\), one can define what it means for a degenerate coset \(e\in{\mathfrak g}_{x,r}/{\mathfrak g}_{x,r+}\) to be distinguished. (Doing so requires partitioning the building \(\mathcal B\) into “generalized \(r\)-facets”. When \(r\) is an integer, these are just facets in the usual sense.) Let \(I_r^d\) denote the set of all distinguished, degenerate Moy-Prasad cosets of depth \(r\). The main theorem of this paper is that, modulo a certain equivalence relation, \(I_r^d\) parametrizes the set of nilpotent orbits in \(\mathfrak g\). Note that for each \(r\in\mathbb{R}\), one thus obtains a parametrization.

The result requires only mild hypotheses on \(k\) and \(G\). These assure the validity of, for example, a graded version of the Jacobson-Morozov Theorem. In particular, no explicit realization of the group \(G\) is required.

A few refinements to the parametrization are contained in joint work of the author and the reviewer [A generalization of a result of Kazhdan and Lusztig, Proc. Am. Math. Soc. (to appear)].

The motivating application of this paper is the author’s proof [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, No. 3, 391-422 (2002; Zbl 0999.22013)] of several homogeneity results for distributions on \(G\) and \(\mathfrak g\), including the Hales-Moy-Prasad Conjecture.

Reviewer: Jeffrey Adler (Akron)