A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

We define the superclasses for a classical finite unipotent group U of type Bn(q), Cn(q), or Dn(q), and show that, together with the supercharacters defined in [C.A.M. André, A.M. Neto, Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups, Pacific J. Math. 239 (2) (2009) 201–230], they form a supercharacter theory in the sense of [P. Diaconis, I.M. Isaacs, Supercharacters and superclasses for algebra groups, Trans. Amer. Math. Soc. 360 (5) (2008) 2359–2392]. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As a consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of U, and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters).