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PROPERTIES OF HIGHER-DIMENSIONAL SHINTANI GENERATING FUNCTIONS AND COCYCLES ON PGL3([Qfr ])

Published online by Cambridge University Press:  12 January 2001

SHUBIN HU  and
DAVID SOLOMON
SHUBIN HU
Affiliation:
2101 Chestnut Street, Room 1508, Philadelphia, PA 19103, USA, [email protected]
DAVID SOLOMON
Affiliation:
Department of Mathematics, King's College, The Strand, London WC2R 2LS, [email protected]
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Abstract

In an earlier paper the second author used the formal, algebraic properties of $2$-dimensional Shintani generating functions to construct a $1$-cocycle on ${\rm PGL}_2(\mathbb{Q})$. We aim to generalise these results by using such functions in dimension $n$ to obtain an $(n-1)$-cocycle on ${\rm PGL}_n(\mathbb{Q})$, presumably related to the Bernoulli and Eisenstein cocycles of R.~Sczech. By improving our methods we achieve this goal for $n=3$. For $n>3$ we encounter obstacles related to degenerate configurations of hyperplanes in $n$-space. Nevertheless, we obtain partial results closely connected to reciprocity laws for certain $n$-dimensional Dedekind sums. 1991 Mathematics Subject Classification: 11F20, 11F75.

Keywords

Dedekind sumsreciprocity law${\rm PGL}_n(\mathbb{Q})$Bernoulli polynomialsEisenstein cocyleShintani's formulae
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 82 , Issue 1 , January 2001 , pp. 64 - 88
Copyright
2001 London Mathematical Society

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