Hecke - Clifford Superalgebras

In [G1], Grojnowski has developed a purely algebraic way to connect the representation theory of affine Hecke algebras at an (ℓ + 1)-th root of unity to the highest weight theory of the affine Kac-Moody algebra of type A (1) ℓ. His work yields amongst other things a new approach to the modular representations of the symmetric groups S n , with the branching rules of [K1, K2, K3, B, BK1] being built into the theory from the outset. For a different approach, not followed here, we also cite the fundamental work of Ariki [A1, A2] and Ariki-Mathas [AM]. The present article is devoted to extending Grojnowski's machinery to the twisted case: we replace the affine Hecke algebras with the affine Hecke-Clifford superalgebras of Jones and Nazarov [JN], and the Kac-Moody algebra A (1) ℓ with the twisted algebra g = A (2) 2ℓ. In particular, we obtain an algebraic construction purely in terms of the representation theory of Hecke-Clifford superalgebras of the plus part U + Z of the enveloping algebra of g, as well as of Kashiwara's highest weight crystals B(∞) and B(λ) for each dominant weight λ. The results of the article have applications to the modular representation theory of the double covers S n of the symmeric groups, as was predicted originally by Leclerc and Thibon [LT], see also [BK2]. In particular, the parametrization of irreducibles, classification of blocks and analogues of the modular branching rules of the symmetric group for the double covers over fields of odd characteristic follow from the special case λ = Λ 0 of our main results. These matters are discussed in the final section of the paper, §9. Let us now describe the main results in more detail. Let H n denote the affine Hecke-Clifford superalgebra of [JN], over an algebraically closed field F of characteristic different from 2 and at defining parameter a primitive (2ℓ + 1)-th root of unity q ∈ F ×. All results also have analogues in the degenerate case q = 1, working instead with the affine Sergeev superalgebra of [N], when the field F should be taken to be of characteristic (2ℓ + 1). We consider K(∞) = n≥0 K(Rep I H n), the sum of the Grothendieck groups of integral Z 2-graded representations of H n for all n (see §4-d for the precise definition). In a familiar way, K(∞) has a natural …