For type SL2-hat, the quantum Bruhat graph formula for the Demazure product on the double affine Weyl semigroup is well-defined for positive levels and associative for levels greater than one.
Affine Deligne-Lusztig varieties associated with generic Newton points
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This paper gives an explicit formula of the dimension of affine Deligne-Lusztig varieties associated with generic Newton point in terms of Demazure product of Iwahori-Weyl groups.
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The Quantum Bruhat Graph for $\widehat{SL}_2$ and Double Affine Demazure Products
For type SL2-hat, the quantum Bruhat graph formula for the Demazure product on the double affine Weyl semigroup is well-defined for positive levels and associative for levels greater than one.