Symmetric subgroup invariants in irreducible representations of \(G^F\), when \(G=\text{GL}_n\)
In the mid 1980s, Lusztig classified the characteristic \(0\) irreducible representations of the fixed point subgroup \(G^F\) of a connected reductive group \(G\), defined over a finite field of odd characteristic, under the Frobenius morphism \(F\). In the paper under review, this is developed in the following way. It is supposed that there is an involution \(\theta\colon G\to G\) of algebraic groups which is assumed to commute with \(F\). Within the fixed-point subgroup \(G^\theta\), we take an \(F\)-stable subgroup \(K\). Then for each irreducible representation \(V\) of \(G^F\), it is desirable to find the dimension of the \(K^F\)-invariants in \(V\). The author achieves this in the case where \(G\) is a general linear group, and \(\theta\) is arbitrary.
scientific article; zbMATH DE number 1901957
scientific article