Geometry of Kottwitz–viehmann Varieties

Abstract

We study basic geometric properties of Kottwitz-Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.Contents 2.2.2. Our approach in this part follows a suggestion of Xinwen Zhu. For any subset J ⊂ ∆, denote, the center of the Levi L J c acts diagonally on the product. There is a canonical mapThe element w ∈ J c W J c corresponds to the G-orbit of (ẇ, 1) for any representativeẇ of w in G. We denote this G-orbit by Y ∅,J,w and let X ∅,J,w be its inverse image under π ∅,J . Then we haveProof. First suppose X ∅,J,w ⊂ N . Then in particularẇe ∅,J ∈ N . Recall that the idempotent e ∅,J acts as projector to highest weight space in the representation V ωi if i ∈ J and acts by 0 if i / ∈ J. If there exists j ∈ J but j / ∈ Supp(w), then ρ ωj (ẇ) preserves the highest weight space in V ωj and hence Tr(ρ ωj (ẇe ∅,J )) = 0, contradiction the assumption thatẇe ∅,J ∈ N .Conversely, then by a standard result in root system we have w(ω i ) = ω i (see, for example [HT06, Lemma 3.5]). Thus we have Tr(ρ ωi (x)) = 0 as t ∈ T preserve the weight spaces andẇ maps the highest weight space into the weight space with weight w(ω i ). Thus x ∈ N . Corollary 2.2.4. (a) There is a stratification of N into Ad(G)-stable pieces N = J⊂∆ w∈ J c W J c Supp(w)⊃J

On affine Lusztig varieties

“…It is a fundamental question to determine the nonemptiness pattern and dimension formula for affine Lusztig varieties. This question was solved for affine Lusztig varieties in the affine Grassmannian of a split connected reductive group in the equal characteristic case (under a mild assumption on the residue characteristic) by Bouthier and Chi [5,6,8]. Their method relies on a global argument using the Hitchin fibration and does not work in the mixed characteristic case (e.g., for Y µ (γ) for a ramified element γ).…”

“…Their method relies on a global argument using the Hitchin fibration and does not work in the mixed characteristic case (e.g., for Y µ (γ) for a ramified element γ). See [8,Remark 1.2.3]. Little is known for nonsplit groups or for the affine flag case.…”