Invariant fields of symplectic and orthogonal groups

Invariant Fields of Symplectic and Orthogonal Groups

The projective orthogonal and symplectic groups POn(F ) and PSpn(F ) have a natural action on the F vector space V ′ = Mn(F ) ⊕ . . . ⊕ Mn(F ). Here we assume F is an infinite field of characteristic not 2. If we assume there is more than one summand in V , then the invariant fields F (V )n and F (V )n are natural objects. They are, for example, the centers of generic algebras with the appropri...

INVARIANT FIELDS OF SYMPLECTIC AND ORTHOGONAL GROUPSDavid

Eigendistributions for Orthogonal Groups and Representations of Symplectic Groups

We consider the action of H = O(p, q) on the matrix space Mp+q,n(R). We study a certain orbit O of H in the null cone N ⊆ Mp+q,n(R) which supports an eigendistribution dνO for H . Using some identities of Capelli type developed in the Appendix, we determine the structure of G̃ = Sp(2n,R)∼-cyclic module generated by dνO under the oscillator representation of G̃ (the metaplectic cover of G = Sp(2n(...

Degree Graphs of Simple Orthogonal and Symplectic Groups

Let G be a finite group and let cd(G) be the set of irreducible ordinary character degrees of G. The degree graph of G is the graph ∆(G) whose set of vertices is the set of primes dividing degrees in cd(G), with an edge between primes p and q if pq divides some degree in cd(G). We determine the graph ∆(G) for the finite simple groups of types B`, C`, D` and D`; that is, for the simple orthogona...

Invariant Theory of Special Orthogonal Groups

In this paper we study the action of SO.n/ on m-tuples of n n matrices by simultaneous conjugation. We show that the polynomial invariants are generated by traces and polarized Pfaffians of skewsymmmetric projections. We also discuss the same problem for other classical groups. 1 Special Orthogonal Groups Let F be a field of characteristic 0. If A is a skewsymmetric 2k 2k matrix over F , we den...