Rapoport's dimension conjecture for affine Deligne–Lusztig varieties

From papers

Let GG be a connected reductive group over FF, let bG(L)b\in G(L), and let μ\mu be a dominant coweight such that the affine Deligne–Lusztig variety Xμ(b)X_\mu(b) is non-empty. Write νˉb\bar\nu_b for the dominant Newton point of bb, let ρ\rho be the half-sum of the positive roots, and let defG(b)\operatorname{def}_G(b) be the FF-rank of GG minus the FF-rank of the inner form JJ of the centralizer group MbM_b. Rapoport's conjecture. The dimension of Xμ(b)X_\mu(b) is

dimXμ(b)=ρ,μνˉb12defG(b).\dim X_\mu(b)=\langle\rho,\mu-\bar\nu_b\rangle-\frac{1}{2}\operatorname{def}_G(b).

This is the expected dimension formula for non-empty affine Deligne–Lusztig varieties in the affine Grassmannian. The source presents it as conjectural and gives no resolution status.

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Sources & referencesView supporting material

Primary source

Ulrich Goertz, Thomas J. Haines, Robert E. Kottwitz and Daniel C. Reuman, “Dimensions of some affine Deligne-Lusztig varieties”, arXiv:math/0504443 (2005).

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