The dimension of the fixed point set on affine flag manifolds
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Let G be a semisimple simply-connected algebraic group over C, g its Lie algebra. Also, F = C((ε)) is the field of formal Laurent series, A = C[[ε]] is the ring of integers in F . Set ĝ = g ⊗ F , gA = g ⊗ A and Ĝ = G(F ). Let B be the set of all Iwahori subalgebras in ĝ, and X the set of all subalgebras in ĝ which are Ĝ-conjugate to gA. Then B and X have the structure of Ind-algebraic varieties over C (they are unions of increasing system of ordinary projective algebraic varieties over C). They are called the affine flag variety and the affine Grassmanian of G respectively. We have X = Ĝ/G(A) and B = Ĝ/I, where I is an Iwahori subgroup. For any N ∈ ĝ let BN ⊂ B (respectively XN ⊂ X) be the set of all Iwahori subalgebras (respectively, subalgebras conjugate to gA) which contain N . Clearly, BN (XN ) is a closed subvariety of the Ind-variety B (respectively X). The varieties BN , XN were studied by Kazhdan and Lusztig in [KL]. Following their paper let us suppose that N is topologically nilpotent (nilelement in the terminology of [KL]), i.e., ad(N) → 0 in EndF (ĝ) when r → ∞. (The topology on EndF (ĝ) arises from the obvious topology on F .) It was shown in loc. cit. that the Ind-varieties BN , XN are finite dimensional iff the element N is regular semisimple. We will assume from now on that this is the case. Then BN and XN are locally finite unions of finite dimensional projective varieties. Moreover, all components of BN have the same dimension, which coincides with the dimension of XN . A precise formula for the dimension of BN was stated in [KL] as a conjecture. The aim of the present note is to give a proof of this conjecture. Let O be the subset of XN defined as follows: if p̂ ∈ XN is a subalgebra conjugate to gA, then p̂ ∈ O iff the image of N in g=̃p̂/εp̂ is a regular nilpotent. Let Z(N) be the centralizer of N in Ĝ; let z(N) be the centralizer of N in ĝ. We also fix a Cartan subalgebra h ⊂ g and denote by W the Weyl group. The result containing the formula for the dimension of BN is the following:
[1] D. Kazhdan,et al. Fixed point varieties on affine flag manifolds , 1988 .
[2] B. Kostant,et al. Lie Group Representations on Polynomial Rings , 1963 .