A GL_d-pseudocharacter is a function from a group Γ to a ring k satisfying polynomial relations that make it “look like” the character of a representation. When k is an algebraically closed field of characteristic 0, Taylor proved that GL_d-pseudocharacters of Γ are the same as degree-d characters of Γ with values in k, hence are in bijection with equivalence classes of semisimple representations Γ → GL_d(k). Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group H over an algebraically closed field k of characteristic 0 and for any group Γ, there exists an infinite collection of functions and relations which are naturally in bijection with H(k)-conjugacy classes of semisimple homomorphisms Γ→ H(k). In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG algebra. We then define generating sets and generating relations for these objects and show that, for all H as above, the corresponding FFG-algebra is finitely presented up to radical. Hence one can always define H-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations up to radical of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of homomorphisms, following Larsen.

GL _d-伪字符是从组Γ到满足多项式关系的环k的函数，该多项式使其“看起来像”表示的字符。当ķ是特征为0的代数闭域，泰勒证明，GL _d Γ的-pseudocharacters的相同度- d与值Γ的字符ķ，因此是在双射半单表示Γ→GL的等价类_d（k）。最近，V. Lafforgue通过展示的是，对于任何连接的还原性基团广义这个结果ħ通过一个代数闭域ķ特征0的任意一个，对于任何群Γ，都存在无限的函数和关系集合，这些函数和关系自然与半简单同态Γ→ H（k）的H（k）共轭类双射。在本文中，我们根据称为FFG代数的新代数对象，对Lafforgue的结果进行了重新表述。然后，我们为这些对象定义生成集并生成关系，并表明，对于上述所有H，相应的FFG代数都有限地表示为根。因此，总是可以定义H-伪字符，由满足有限多个关系的有限多个函数组成。接下来，我们使用不变理论对（一般）正交群，（一般）辛群和特殊正交群的FFG代数的根给出明确的有限表示。最后，在拉森之后，我们使用伪字符回答有关同态的共轭与元素共轭的问题。