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PSEUDOCHARACTERS OF HOMOMORPHISMS INTO CLASSICAL GROUPS
Transformation Groups ( IF 0.4 ) Pub Date : 2020-08-06 , DOI: 10.1007/s00031-020-09603-2
M. WEIDNER

A GLd-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 GLd-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 → GLd(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.



中文翻译:

同质化为经典群的伪特征

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dk)。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→ Hk)的Hk)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。

更新日期:2020-08-06
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