Arithmetic Study Group: A generalized Weil representation for the finite split ortogonal group O_q(2n,2n), q odd and greater than 3.
3 November 2015 14:00 in CM 103
Weil representations have proven to be a powerful tool in the theory of group representations. They originate from a very general construction of A. Weil, which has as a consequence the existence of a projective representation of the group Sp(2n,K), K a locally compact field. In particular, these representations have allowed to build all irreducible complex linear representations of the general linear group of rank 2 over a finite field, and later over a local field, except in residual characteristic two.
I will start by giving a introduction to Weil representations and some examples for classical groups, like SL(2,k), k a finite field.
Then we will define SL*(2,A) groups, which are an analogue of SL(2,k) but with entries over an involutive ring A. When these groups have a Bruhat-like presentation, Gutierrez, Soto-Andrade and Pantoja have developed a method for constructing generalized Weil representations for them.
I will construct a generalized Weil representation for the finite split orthogonal group O(2n,2n), seen as a SL*-group. Furthermore, we will see that these representation is equal to the restriction of the Weil representation to O(2n,2n) for the dual pair (Sp(2,k), O(2n,2n)). This fact shows an example of compatibility between the mentioned method with “classical” methods.
If time allows we will discuss about an initial decomposition.
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