Download Operator Algebras: The Abel Symposium 2004 (Abel Symposia) by Editors: Ola Bratteli, Sergey Neshveyev, and Christian Skau PDF

By Editors: Ola Bratteli, Sergey Neshveyev, and Christian Skau

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The most function of those lectures is first to in short survey the basic con­ nection among the illustration concept of the symmetric staff Sn and the idea of symmetric services and moment to teach how combinatorial tools that come up certainly within the concept of symmetric services result in effective algorithms to specific numerous prod­ ucts of representations of Sn when it comes to sums of irreducible representations.

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Consider the two Q-lattices (α1 , z1 ) = (0, 2 −1) and (α2 , z2 ) = (0, −1). The composite ((α1 , z1 ), (α2 , z2 )) ◦ ((α2 , z2 ), (α1 , z1 )) is equal to the identity ((α1 , z1 ), (α1 , z1 )). We can also consider the composition √ √ ( −1 (α1 , z1 ), −1 (α2 , z2 )) ◦ ((α2 , z2 ), (α1 , z1 )), √ √ where −1 (α2 , z2 ) = (α2 , z2 ), but this is not the identity, since −1 (α1 , z1 ) = (α1 , z1 ). However, we can still define a convolution algebra for the quotient Z of (50), by restricting the convolution product of R2 to homogeneous functions of weight zero with C∗ -compact support, where a function f has weight k if it satisfies f (g, α, uλ) = λk f (g, α, u), ∀λ ∈ C∗ .

1 Tower Power If V is an algebraic variety – or a scheme or a stack – over a field k, a “tower” T over V is a family Vi (i ∈ I) of finite (possibly branched) covers of V such that for any i, j ∈ I, there is a l ∈ I with Vl a cover of Vi and Vj . Thus, I is a partially ordered set. This gives a corresponding compatible system of covering maps Vi → V . In case of a tower over a pointed variety (V, v), one fixes a point vi over v in each Vi . Even though Vi may not be irreducible, we shall allow ourselves to loosely refer to Vi as a variety.

The refined theory of automorphisms of the modular field does not enter in any way in the definition of the algebra. ˆ let Gα ⊂ GL+ (Q) be the set of For α ∈ M2 (Z), 2 ˆ Gα = {g ∈ GL+ 2 (Q) : gα ∈ M2 (Z)}. 40 Alain Connes, Matilde Marcolli, and Niranjan Ramachandran ˆ × H determines a Then, as shown in [10], an element y = (α, z) ∈ M2 (Z) unitary representation of the Hecke algebra A on the Hilbert space 2 (Γ \Gα ), f (gs−1 , sα, s(z)) ξ(s), ((πy f )ξ)(g) := ∀g ∈ Gα (63) s∈Γ \Gα for f ∈ A and ξ ∈ 2 (Γ \Gα ).

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