Group Actions of Some Subgroups of Parabolic Subgroups

Abstract

LetKbe the finite field withqelements,Vann-dimensional vector space overK,Pthe parabolic subgroup ofGL(V) associated to the flagV≔V0⊃V1⊃…⊃Vr={0},Uthe unipotent radical ofP. Suppose thatGis a subgroup ofGL(V) such that (i)U⊂G⊂P, and (ii) Image {G→GL(V0/V1)×GL(V1/V2)×…×GL(Vr−1/Vr)}=G1×G2×…×Gr, whereGiis a subgroup ofGL(Vi−1/Vi) for 1≤i≤r.THEOREM(i)If K[Vi−1/Vi]Giis a polynomial ring(resp. a complete intersection, a Gorenstein ring, a Cohen–Macaulay ring)for1≤i≤r,so is K[V]G. (ii)If K[Vi−1/Vi]is a free module over K[Vi−1/Vi]Gifor1≤i≤r,so is K[V]over K[V]G. (iii)If K(Vi−1/Vi)Giis rational over K for1≤i≤r,so is K(V)Gover K.