On \(G_2(\mathbb{F}_q)\)-invariants of representations of \(D_4(\mathbb{F}_q)\). (English) Zbl 1200.20032
Summary: Let \(\mathbb{F}_q\) be a finite field of order \(q\) and whose characteristic is not equal to 2 or 3. Let \(G\) denote a split algebraic group of type \(D_4\) over \(\mathbb{F}_q\). It contains a split algebraic group \(G_2\) over \(\mathbb{F}_q\). In this paper we state a conjecture on the dimensions of \(G_2(\mathbb{F}_q)\)-invariant vectors on representations of \(G(\mathbb{F}_q)\), and we prove this conjecture in the cases where the representation of \(G(\mathbb{F}_q)\) is a principal series representation, a generic representation, or a unipotent representation.
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20C33 | Representations of finite groups of Lie type |
| 20G40 | Linear algebraic groups over finite fields |
Keywords:
finite groups of Lie type; principal series representations; generic representations; unipotent representations; orbitsReferences:
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