Representations of Hecke Algebras at Roots of Unity by Meinolf Geck

By Meinolf Geck

The modular illustration concept of Iwahori-Hecke algebras and this theory's connection to teams of Lie variety is a space of swiftly increasing curiosity; it truly is one who has additionally obvious a few breakthroughs lately. In classifying the irreducible representations of Iwahori-Hecke algebras at roots of solidarity, this ebook is a very useful addition to present learn during this box. utilizing the framework supplied through the Kazhdan-Lusztig thought of cells, the authors enhance an analogue of James' (1970) "characteristic-free'' method of the illustration conception of Iwahori-Hecke algebras in general.

Presenting a scientific and unified therapy of representations of Hecke algebras at roots of cohesion, this e-book is exclusive in its strategy and contains new effects that experience no longer but been released in ebook shape. It additionally serves as historical past interpreting to extra energetic components of present examine reminiscent of the speculation of affine Hecke algebras and Cherednik algebras.

The major result of this ebook are bought by means of an interplay of a number of branches of arithmetic, particularly the idea of Fock areas for quantum affine Lie algebras and Ariki's theorem, the combinatorics of crystal bases, the idea of Kazhdan-Lusztig bases and cells, and computational methods.

This ebook should be of use to researchers and graduate scholars in illustration idea in addition to any researchers outdoor of the sector with an curiosity in Hecke algebras.

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We have the following relations: γ˜x,y,z = γ˜y,z,x (a) ∑ γ˜x−1 ,y,w n˜w = δxy (b) for all x, y, z ∈ W , for all x, y ∈ W , w∈W n˜w = n˜ w−1 and γ˜x,y,z = γ˜y−1 ,x−1 ,z−1 (c) for all w, x, y, z ∈ W . Proof. (a) Just note that the defining formula for γ˜x,y,z is symmetrical under cyclic permutations of x, y, z. (b) Using the defining formulae for γ˜x,y,z and n˜ w , the left-hand side evaluates to ∑ ∑ w∈W ∑ λ ∈Λ s,t,u∈M(λ ) = ∑ tu us fλ−1 cst x−1 ,λ cy,λ cw,λ ∑ ∑ λ ,μ ∈Λ s,t,u∈M(λ ) v∈M(μ ) ∑ ∑ μ ∈Λ v∈M(μ ) tu f λ−1 f μ−1 cst x−1 ,λ cy,λ f μ−1 cvv w−1 , μ vv ∑ cus w,λ cw−1 ,μ .

Then we perform a slight transformation as follows. We set σˆ εj (Tw ) := P−1 σ εj (Tw )P (w ∈ W ), where P := 1 − 12 0 1 . Then Ωˆ j σˆ εj (Tw−1 ) = σˆ εj (Tw )tr Ωˆ j for all w ∈ W , where Ωˆ j := Ptr Ω j P. Furthermore, 24 1 Generic Iwahori–Hecke Algebras Ωˆ j ≡ 20 0 12 Ωˆ j ≡ 0 2(2 + ζ j + ζ − j ) 1 j −j 0 2 (2 − ζ − ζ ) if L(s1 ) > L(s2 ) = 0, mod m if L(s1 ) = L(s2 ) = 0. 5 applies again and so σˆ εj is balanced. But then σ εj must also be balanced, since the transforming matrix P has all its entries in K.

From the explicit knowledge of cλ one can deduce explicit formulae for the invariants aλ and f λ . If L(s) = 0 for all s ∈ S, then cλ = |W |/dλ . Hence, aλ = 0 and fλ = |W |/dλ for all λ ∈ Λ in this case. Now assume that L(s) > 0 for at least some s ∈ S. For W of exceptional type H3 , H4 , E6 , E7 , E8 (where we are automatically in the equal-parameter case), see the tables in [220, Chap. 4] and in the Appendices C and E in [132]. 2 (p. 3 Lusztig’s a-Invariants 17 diagram, we can assume without loss generality that L(s1 ) = L(s2 ) For the types I2 (m), An−1 , Bn and Dn , see the examples below.

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