Multiplicity one theorems for GSp(k,2n) and O(k,n), where k is a finite field
Let G be a conformal symplectic group GSp(2n,k), where k is a finite field with char(k)\(\neq 2\), and let Q be a parabolic subgroup of G whose Levi subgroup is of type \(C_{n-1}\). Fix a Levi decomposition \(Q=MU\), where M is a Levi subgroup of Q and U is the unipotent radical of Q. Then U is isomorphic to the Heisenberg group of order 2n-2, and we denote by \(M_ 1\) the centralizer in M of the center of U. One can construct the Weil representation \(\chi\) on \(D=M_ 1U\) with respect to the Heisenberg group U. In this paper, the author proves that the induced representation \(\rho =Ind^ G_ D\chi\) is multiplicity free. For this, it is enough to show that the endomorphism algebra \(A=End_ G\rho\) is commutative. She proves this by showing that a certain involutive antihomomorphism \(\tau\) : \(A\to A\) is in fact the identity. She also considers, as an analogue to the case above, a certain induced representation \(\rho\) in the case of orthogonal groups \(G=O(n,k)\), and shows that \(\rho\) is ``almost'' multiplicity free, in the sense that \(\tau\) : \(A\to A\) (in a setting similar to that in the case of GSp(2n,k)) is almost the identity.
scientific article; zbMATH DE number 4116810
scientific article