Dimensions of Newton strata in the adjoint quotient of reductive groups.

*(English)*Zbl 1109.11033The author interprets the classical theory of Newton polygons in a group theoretic way (the underlying group being the general linear group), and extends it to other reductive groups. This gives rise to the Newton strata mentioned in the title, and the dimensions of these strata are determined.

More precisely, consider monic polynomials with non-zero constant term, with coefficients in a complete discretely valued field \(F\). The classical theory of Newton polygons says that the valuations of the roots of the polynomial are determined by the valuations of the coefficients, and also which coefficient valuations can occur if the valuations of the roots are fixed.

If \(A \subset \text{GL}_n\) is the diagonal torus, and \(W\) is its Weyl group, then we can identify \(\mathbb A(F) := (A/W)(F)\) with the set of polynomials as above, mapping a polynomial to the unordered tuple of its roots. Mapping a polynomial to the tuple of valuations of its coefficients, we obtain a map \(\Phi\colon\mathbb A(F) \rightarrow \mathcal N_{\text{GL}_n}\), where \(\mathcal N_{\text{GL}_n} \subset X_*(A)_{\mathbb Q,\text{dom}}\) is the set of tuples occurring as slopes of a Newton polygon.

More generally, the author starts with any split connected reductive group \(G\) over the ring of integers of \(F\) (s.t. the derived group of \(G\) is simply connected). Fix a Borel subgroup and a maximal torus \(A\). We then again have the notions of \(\mathbb A = A/W\), and of \(\mathcal N_G \subset X_*(A)_{\mathbb Q,\text{dom}}\). Using the “closest point map” which retracts \(X_*(A)_{\mathbb R}\) onto \(X_*(A)_{\mathbb R, \text{dom}}\), the author defines a map \(\mathbb A(F) \rightarrow \mathcal N_{G}\). The fibers \(\mathbb A(F)_\mu\) (\(\mu \in \mathcal N_G\)) of this map are called Newton strata in the adjoint quotient.

The author determines the dimensions of the unions \(\mathbb A(F)_{\leq \mu} = \bigcup_{\nu \leq \mu} \mathbb A(F)_\nu\), and goes on to rewrite the dimension formula in a simpler form using the notion of defect \(\text{def}_G(\nu)\in\mathbb Z\), a group-theoretic invariant associated with \(\nu\) in a simple and explicit way.

Interestingly, the codimension of \(\mathbb A(F)_{\leq \nu}\) in \(\mathbb A(F)_\mu\) coincides with the value predicted by Chai for the codimension of the Newton stratum associated with \(\nu\) in any Shimura variety with associated minuscule coweight \(\mu\) (where \(G\) splits over \(\mathbb Q_p\)) [see C.-L. Chai, Am. J. Math. 122, No. 5, 967–990 (2000; Zbl 1057.11506)]. It is also related to the formula for dimensions of affine Deligne-Lusztig varieties in the affine Grassmannian [see U. Görtz; T. J. Haines, R. E. Kottwitz and D. C. Reuman, Ann. Sci. Éc. Norm. Supér. (4) 39, No. 3, 467–511 (2006; Zbl 1108.14035)] and the references given there).

More precisely, consider monic polynomials with non-zero constant term, with coefficients in a complete discretely valued field \(F\). The classical theory of Newton polygons says that the valuations of the roots of the polynomial are determined by the valuations of the coefficients, and also which coefficient valuations can occur if the valuations of the roots are fixed.

If \(A \subset \text{GL}_n\) is the diagonal torus, and \(W\) is its Weyl group, then we can identify \(\mathbb A(F) := (A/W)(F)\) with the set of polynomials as above, mapping a polynomial to the unordered tuple of its roots. Mapping a polynomial to the tuple of valuations of its coefficients, we obtain a map \(\Phi\colon\mathbb A(F) \rightarrow \mathcal N_{\text{GL}_n}\), where \(\mathcal N_{\text{GL}_n} \subset X_*(A)_{\mathbb Q,\text{dom}}\) is the set of tuples occurring as slopes of a Newton polygon.

More generally, the author starts with any split connected reductive group \(G\) over the ring of integers of \(F\) (s.t. the derived group of \(G\) is simply connected). Fix a Borel subgroup and a maximal torus \(A\). We then again have the notions of \(\mathbb A = A/W\), and of \(\mathcal N_G \subset X_*(A)_{\mathbb Q,\text{dom}}\). Using the “closest point map” which retracts \(X_*(A)_{\mathbb R}\) onto \(X_*(A)_{\mathbb R, \text{dom}}\), the author defines a map \(\mathbb A(F) \rightarrow \mathcal N_{G}\). The fibers \(\mathbb A(F)_\mu\) (\(\mu \in \mathcal N_G\)) of this map are called Newton strata in the adjoint quotient.

The author determines the dimensions of the unions \(\mathbb A(F)_{\leq \mu} = \bigcup_{\nu \leq \mu} \mathbb A(F)_\nu\), and goes on to rewrite the dimension formula in a simpler form using the notion of defect \(\text{def}_G(\nu)\in\mathbb Z\), a group-theoretic invariant associated with \(\nu\) in a simple and explicit way.

Interestingly, the codimension of \(\mathbb A(F)_{\leq \nu}\) in \(\mathbb A(F)_\mu\) coincides with the value predicted by Chai for the codimension of the Newton stratum associated with \(\nu\) in any Shimura variety with associated minuscule coweight \(\mu\) (where \(G\) splits over \(\mathbb Q_p\)) [see C.-L. Chai, Am. J. Math. 122, No. 5, 967–990 (2000; Zbl 1057.11506)]. It is also related to the formula for dimensions of affine Deligne-Lusztig varieties in the affine Grassmannian [see U. Görtz; T. J. Haines, R. E. Kottwitz and D. C. Reuman, Ann. Sci. Éc. Norm. Supér. (4) 39, No. 3, 467–511 (2006; Zbl 1108.14035)] and the references given there).

Reviewer: Ulrich Görtz (Bonn)