Abstract
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.
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M. WEIDNER is partially supported by Caltech’s Samuel P. and Frances Krown SURF Fellowship and a Churchill Scholarship from the Winston Churchill Foundation of the USA.
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WEIDNER, M. PSEUDOCHARACTERS OF HOMOMORPHISMS INTO CLASSICAL GROUPS. Transformation Groups 25, 1345–1370 (2020). https://doi.org/10.1007/s00031-020-09603-2
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DOI: https://doi.org/10.1007/s00031-020-09603-2