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

Easy difficulties of illustration conception are to categorise irreducible representations and decompose representations occuring evidently in another context. Algebras of Iwahori-Hecke sort are one of many instruments and have been, most likely, first thought of within the context of illustration concept of finite teams of Lie sort. This quantity comprises notes of the classes on Iwahori-Hecke algebras and their illustration thought, given in the course of the CIME summer time tuition which happened in 1999 in Martina Franca, Italy.

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**Extra resources 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**

This induces an antiautomorphism of the convolution algebra Cc∞ (G) and is the identity upon restriction to H. A similar argument works for Sp2n as long as we use coordinates with respect to a symplectic basis. Thus irreducible representations of K occur at most once in any representation of G. The operators originally constructed by Hecke were elements of H(G//K) where G = SL2 or G = PGL2 . 2 H(G//J ) The extended aﬃne Weyl group For G = GLn , the Weyl group W is the group of permutations, Sn , generated by the n − 1 transpositions sj for j = 1, .

We know from the Bruhat theory for GL(V ) 40 Roger Howe (Lecture Notes by Cathy Kriloﬀ) that we can ﬁnd a line decomposition of V compatible with both F1 and F2 . However, this does us little good in understanding the structure of Sp(V ), because Sp(V ) does not act transitively on the set of line decompositions of V . In the context of Sp(V ), we need to capitalize on the self-dual structure of the Fj to show that we can ﬁnd a line decomposition compatible with the symplectic structure. Let V = ⊕j Lj be a line decomposition of V .

The groups KL are referred to as parahoric subgroups. If L is complete, we also write KL = J. This is the Iwahori subgroup of Sp(V ). If K ⊂ Sp(V ) is any compact subgroup, then as we have seen in the discussion of GL(V ), K will preserve some lattice Λ in V . It will then also preserve all lattices obtained from L by the operations of scalar multiplication, duality, intersection and sum. In other words, it will preserve the self-dual lattice of lattices generated by L. 5, it will preserve some almost self-dual lattice, and likewise the self-dual lattice ﬂag it generates.