PSEUDOCHARACTERS OF HOMOMORPHISMS INTO CLASSICAL GROUPS
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Springer Science+Business Media New York (2020)
PSEUDOCHARACTERS OF HOMOMORPHISMS INTO CLASSICAL GROUPS M. WEIDNER∗ Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213, USA [email protected]
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.
Introduction Pseudocharacters were originally introduced for GL2 by Wiles [Wil] and generalized to GLn by Taylor [Tay]. Taylor’s result on GLn -pseudocharacters is as follows. Let Γ be a group and k be a commutative ring with identity. Define a GLn pseudocharacter of Γ over k to be a set map T : Γ → k such that • T (1) = n, • For all γ1 , γ2 ∈ Γ, T (γ1 γ2 ) = T (γ2 γ1 ), • For all γ1 , . . . , γn+1 ∈ Γ, X sgn(σ)Tσ (γ1 , . . . , γn+1 ) = 0, (1) σ∈Sn+1
DOI: 10.1007/S00031-020-09603-2 Partially supported by Caltech’s Samuel P. and Frances Krown SURF Fellowship and a Churchill Scholarship from the Winston Churchill Foundation of the USA. Received September 4, 2018. Accepted March 28, 2020. Corresponding Author: M. Weidner, e-mail: [email protected] ∗
M. WEIDNER
where Sn+1 is the symmetric group on n+1 letters, sgn(σ) is the permutation sign of σ, and Tσ is defined by Tσ (γ1 , . . . , γn+1 ) = T (γi(1) · · · γi(1) ) · · · T (γi(s) · · · γi(s) ) 1
(1)
r1
(1)
rs
1
(s)
(s)
where σ has cycle decomposition (i1 . . . ir1 ) · · · (i1 . . . irs ). If T is a GLn -pseudocharacter, then define the kernel of T by ker(T ) = {η ∈ Γ | T (γη) = T (γ) for all γ ∈ Γ}. Then: Theorem 1 ([Tay, Thm. 1]). (1) Let ρ : Γ → GLn (k) be a representation. Then tr(ρ) is a GLn -pseudocharacter. (2) Suppose k is a field of characteristic 0, and let
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