For any Kac–Moody group , we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for is strictly compatible with a -valued length function. We conjecture in general and prove for of affine ADE type that the Bruhat order is graded by this length function. We also formulate and discuss conjectures relating the length function to intersections of “double-affine Schubert varieties”.
Revised: 2018-05-24
Accepted: 2018-07-23
Published online: 2019-03-05
DOI: https://doi.org/10.5802/alco.37
Classification: 05E10
Keywords: Kac–Moody groups, double-affine Bruhat order
@article{ALCO_2019__2_2_197_0, author = {Muthiah, Dinakar and Orr, Daniel}, title = {On the double-affine Bruhat order: the $\varepsilon =1$ conjecture and classification of covers in ADE type}, journal = {Algebraic Combinatorics}, pages = {197--216}, publisher = {MathOA foundation}, volume = {2}, number = {2}, year = {2019}, doi = {10.5802/alco.37}, mrnumber = {3934828}, zbl = {1414.05304}, language = {en}, url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_2_197_0/} }
Muthiah, Dinakar; Orr, Daniel. On the double-affine Bruhat order: the $\varepsilon =1$ conjecture and classification of covers in ADE type. Algebraic Combinatorics, Volume 2 (2019) no. 2, pp. 197-216. doi : 10.5802/alco.37. https://alco.centre-mersenne.org/item/ALCO_2019__2_2_197_0/
[1] Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Volume 231, Springer, 2005, xiv+363 pages | MR 2133266 | Zbl 1110.05001
[2] Pursuing the double affine Grassmannian. I. Transversal slices via instantons on -singularities, Duke Math. J., Volume 152 (2010) no. 2, pp. 175-206 | Article | MR 2656088 | Zbl 1200.14083
[3] Iwahori-Hecke algebras for -adic loop groups, Invent. Math., Volume 204 (2016) no. 2, pp. 347-442 | Article | MR 3489701 | Zbl 1345.22011
[4] Semi-infinite flags. I. Case of global curve , Differential topology, infinite-dimensional Lie algebras, and applications (Advances in the Mathematical Sciences) Volume 194, American Mathematical Society, 1999, pp. 81-112 | Article | MR 1729360 | Zbl 1076.14512
[5] Infinite-dimensional Lie algebras, Cambridge University Press, 1990, xxii+400 pages | Article | MR 1104219 | Zbl 0716.17022
[6] On Iwahori-Hecke algebras for -adic loop groups: double coset basis and Bruhat order, Am. J. Math., Volume 140 (2018) no. 1, pp. 221-244 | Article | MR 3749194 | Zbl 1390.22019