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On Iwahori-Hecke algebras for $$p$$-adic loop groups: double coset basis and Bruhat order. (English) Zbl 1390.22019
This paper studies the $$p$$-adic loop group Iwahori-Hecke algebra. The first main result gives an algebraic construction of the double coset basis of this algebra given by indicator functions of double cosets. In particular, it is shown that the structure coefficients with respect to the double coset basis are polynomials in the order of the residue field. The elements of this basis are indexed by elements of a certain semigroup (the semidirect product of a Weyl group and the Tits cone). There is a natural analogue of the Bruhat order on this semigroup which is, a priori, only a preorder. The second main result of the paper asserts that this preorder is, in fact, a partial order. For this latter fact, a non-trivial generalization of the length function is defined. This function takes values in $$\mathbb Z\oplus\mathbb Z$$ considered as an ordered set using the lexicographic order.

##### MSC:
 22E67 Loop groups and related constructions, group-theoretic treatment 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 20G44 Kac-Moody groups 20C08 Hecke algebras and their representations
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