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Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p , q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p-1}{q-1}$$ q p - 1 q - 1 , then every non-scalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.
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Global B(G) with adelic coefficients and transfer factors at non-regular elements
The goal of this paper is extend Kottwitz’s theory of B(G) for global fields. In particular, we show how to extend the definition of “B(G) with adelic coefficients” from tori to all connected reductive groups. As an application, we give an explicit construction of certain transfer factors for non-regular semisimple elements of non-quasisplit groups. This generalizes some results of Kaletha and Taibi. These formulas are used in the stabilization of the cohomology of Shimura and Igusa varieties.
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- Award ID(s):
- 1840234
- PAR ID:
- 10531525
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Mathematische Zeitschrift
- Volume:
- 306
- Issue:
- 4
- ISSN:
- 0025-5874
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00209-024-03454-3
