Dimensions of some affine Deligne–Lusztig varieties

By Ulrich Goertz, Thomas J. Haines, R. Kottwitz, Daniel C. Reuman. 2005

97 citations recorded by the source corpus.

This paper concerns the dimensions of certain affine Deligne–Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. Rapoport conjectured a formula for the dimensions of the varieties Xμ(b) in the affine Grassmannian. We prove his conjecture for b in the split torus; we find that these varieties are equidimensional; and we reduce the general conjecture to the case of superbasic b. In the affine flag manifold, we prove a formula that reduces the dimension question for Xx(b) with b in the split torus to computations of dimensions of intersections of Iwahori orbits with orbits of the unipotent radical. Calculations using this formula allow us to verify a conjecture of Reuman in many new cases, and to make progress toward a generalization of his conjecture.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d9d95f32d332033f60bdd906b892d50280f4d09a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Xuhua He
2. U. Hartl
3. E. Viehmann
4. S. Nie
5. Michael Rapoport
6. Ulrich Goertz

In Geometry and Topology

1. Xuhua He
2. E. Viehmann
3. U. Hartl
4. Michael Rapoport
5. Peter Scholze
6. Xinwen Zhu
7. T. Wedhorn
8. Georgios Pappas

In Mathematical Physics

1. Xuhua He
2. E. Viehmann
3. U. Hartl
4. Michael Rapoport
5. Peter Scholze
6. Xinwen Zhu
7. T. Wedhorn
8. Georgios Pappas

Citations in the corpus

• GEOMETRIC AND HOMOLOGICAL PROPERTIES OF AFFINE DELIGNE-LUSZTIG VARIETIES (2012, methodology): “This is achieved by combining the “dimension=degree” theorem with the partial conjugation method developed in [13] and the dimension formula for affine Deligne-Lusztig varieties in the affine Grassmannian in [7] and [31].”
• The dimension of some affine Deligne–Lusztig varieties (2006, methodology): “We recall the definitions of affine Deligne–Lusztig varieties from [6,1].”
• The Newton stratification on deformations of local G-shtukas (2008, methodology): “2 has been shown in [16], Proposition 2.”
• Towards a theory of local Shimura varieties (2014, unknown): “[21, 26, 28, 35, 79].”