By Ivan Cherednik, Yavor Markov, Roger Howe, George Lusztig, Dan Barbasch, M. Welleda Baldoni

ISBN-10: 3540002243

ISBN-13: 9783540002246

Easy difficulties of illustration conception are to categorise irreducible representations and decompose representations occuring evidently in another context. Algebras of Iwahori-Hecke variety are one of many instruments and have been, most likely, first thought of within the context of illustration idea of finite teams of Lie style. This quantity includes notes of the classes on Iwahori-Hecke algebras and their illustration thought, given through the CIME summer season university which came about in 1999 in Martina Franca, Italy.

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**Additional info for Iwahori-Hecke algebras and their representation theory: lectures given at the C.I.M.E. summer school held in Martina Franca, Italy, June 28-July 6, 1999**

**Example text**

2. GL(V ) = U W AU = U AW U = U w AW U w , for any w, w ∈ W , where U w = wU w−1 . This could also be written as G = U W U where the aﬃne Weyl group, W = W A is the semidirect product of W and A, or the stabilizer of the overall line decomposition of V . The Bruhat Decomposition also has the following equivalent more geometric reformulation. 3. G = BWB if and only if for any two ﬂags, F1 and F2 , there exists a line decomposition of V compatible with both F1 and F2 . Proof. First suppose that the geometric version holds.

Hence it is the multiplication by a constant thanks to the irreducibility of V2n+1 . The constant is F− ◦ F+ (1) and can be readily calculated. It is equally simple to calculate all F− ◦ F+ (xl ) using (50) and (52). For instance, 24 Ivan Cherednik and Yavor Markov (−1)m m! 2n−2m λ = (n − m)! (−1)m m! (−1)n−m (n − m)! 2m = x = (−1)n x2m . (n − m)! m! F− ◦ F+ (x2m ) = F− Thus the truncated inversion reads: F− ◦ F+ = (−1)n id = F+ ◦ F− . Concerning the Plancherel formula, we may use the proportionality of the forms f, g + and f , g − for f, g ∈ V2n+1 and their transforms f = F(f ), g = F(g).

Let {ej , fj } be a symplectic basis for V . If Λo is the lattice spanned by this basis, then it is easy to see that Λ∗o = Λo . We say that Λo is self dual. For 1 ≤ a ≤ n (where dim V = 2n), let Λa be the lattice spanned by {ej , 1 ≤ j ≤ n} ∪ {fj , a < j ≤ n} ∪ {πfj , 1 ≤ j ≤ a}. Let Λn+a be the lattice spanned by {ej , 1 ≤ j ≤ n − a} ∪ {πej , n − a < j ≤ n} ∪ {πfj , 1 ≤ j ≤ n}. Then set Λb+2nc = π c Λb for 0 ≤ b < 2n, and c ∈ Z. The reader can check that the Λm form a complete lattice ﬂag, which is self-dual.

### Iwahori-Hecke algebras and their representation theory: lectures given at the C.I.M.E. summer school held in Martina Franca, Italy, June 28-July 6, 1999 by Ivan Cherednik, Yavor Markov, Roger Howe, George Lusztig, Dan Barbasch, M. Welleda Baldoni

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