By Garrett P.

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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.

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