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Recall that Z= T G=B NT G=Bis a union of lagrangians in T G=B T G=B indexed by the Weyl group. Any class 2H G (Z) (thought of as equivariant Borel-Moore homology) de nes an endomorphism of H G (T G=B) by ( ) = (ˇ 2) \ˇ 1 : As before, these endomorphisms form an algebra under convolution. Recall that

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Example 1.3 : Let T denote the unit circle. Consider the binary operation of multiplication from T T into T:Note that the action (1.1) (z;w) ! zw can be interpreted as rotation of zabout the origin through the angle arg(w) in the anticlockwise direction. As an another interesting example of a binary operation, consider the

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Recall that Z= T G=B NT G=Bis a union of lagrangians in T G=B T G=B indexed by the Weyl group. Any class 2H G (Z) (thought of as equivariant Borel-Moore homology) de nes an endomorphism of H G (T G=B) by ( ) = (ˇ 2) \ˇ 1 : As before, these endomorphisms form an algebra under convolution. Recall thatThomas Gimbert Booth, born in June 1850, was the son of Thomas Booth who was a pottery manufacturer, of. Knowles Works, Burslem in 1864 and of Church Bank Works about 1870. Thomas Gimbert Booth succeeded his father about 1876 and for a short time was in partnership with his brother Fred as T G & F Booth.

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1. (a ∗b) ∗c = a ∗(b ∗c) ∀a,b,c ∈S (associativity), 2. There exists an element e ∈G such that e ∗a = a and a ∗e = a ∀a ∈G (identity), 3. For each a ∈G, there exists an element b ∈G such that a ∗b = e and b ∗a = e (inverse). Definition 1.1.3. A group (G,∗) is called abelian (or commutative) if a ∗b = b ∗a ∀ ...

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