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For any integer $n geq 2,$ we may identify the space of unimodular lattices in $mathbb{R}^n$ with the homogeneous space $X_n := mathrm{SL}_n(mathbb{R})/mathrm{SL}_n(mathbb{Z})$ via the identification $g mathbb{Z}^n = gmathrm{SL}_n(mathbb{Z}).$ Given any $L^1$ function $f : mathbb{R}^n to mathbb{R},$ we define its Siegel transform $widehat{f} : X_n to mathbb{R}$ by $widehat{f}(Lambda) = sum_{v in Lambda - {0}} f(v).$ The Siegel Mean Value Theorem then asserts that the Siegel transform defines an $L^1$ isometry. (The measure on $X_n$ is the probability measure induced in the usual fashion from the Haar measure on $mathrm{SL}_n(mathbb{R}).)$

I am aware that Weil proved an analogue of the above theorem in the case of number fields and that analogues also exist in the space of real symplectic lattices. My question is whether there exist any analogues of Siegel's Theorem for other homogeneous spaces. I would also be grateful if someone could provide me with a reference for proofs of the Siegel Mean Value Theorem that are written in notation and style more contemporary than Siegel's original paper.

Thank you!

number-theory reference-request harmonic-analysis lattices-in-lie-groups arithmetic-dynamics

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asked 2 days ago

Mishel SkenderiMishel Skenderi

1699

$endgroup$

add a comment | 

0

$begingroup$

For any integer $n geq 2,$ we may identify the space of unimodular lattices in $mathbb{R}^n$ with the homogeneous space $X_n := mathrm{SL}_n(mathbb{R})/mathrm{SL}_n(mathbb{Z})$ via the identification $g mathbb{Z}^n = gmathrm{SL}_n(mathbb{Z}).$ Given any $L^1$ function $f : mathbb{R}^n to mathbb{R},$ we define its Siegel transform $widehat{f} : X_n to mathbb{R}$ by $widehat{f}(Lambda) = sum_{v in Lambda - {0}} f(v).$ The Siegel Mean Value Theorem then asserts that the Siegel transform defines an $L^1$ isometry. (The measure on $X_n$ is the probability measure induced in the usual fashion from the Haar measure on $mathrm{SL}_n(mathbb{R}).)$

I am aware that Weil proved an analogue of the above theorem in the case of number fields and that analogues also exist in the space of real symplectic lattices. My question is whether there exist any analogues of Siegel's Theorem for other homogeneous spaces. I would also be grateful if someone could provide me with a reference for proofs of the Siegel Mean Value Theorem that are written in notation and style more contemporary than Siegel's original paper.

Thank you!

number-theory reference-request harmonic-analysis lattices-in-lie-groups arithmetic-dynamics

share|cite|improve this question

asked 2 days ago

Mishel SkenderiMishel Skenderi

1699

$endgroup$

add a comment | 

$begingroup$

For any integer $n geq 2,$ we may identify the space of unimodular lattices in $mathbb{R}^n$ with the homogeneous space $X_n := mathrm{SL}_n(mathbb{R})/mathrm{SL}_n(mathbb{Z})$ via the identification $g mathbb{Z}^n = gmathrm{SL}_n(mathbb{Z}).$ Given any $L^1$ function $f : mathbb{R}^n to mathbb{R},$ we define its Siegel transform $widehat{f} : X_n to mathbb{R}$ by $widehat{f}(Lambda) = sum_{v in Lambda - {0}} f(v).$ The Siegel Mean Value Theorem then asserts that the Siegel transform defines an $L^1$ isometry. (The measure on $X_n$ is the probability measure induced in the usual fashion from the Haar measure on $mathrm{SL}_n(mathbb{R}).)$

I am aware that Weil proved an analogue of the above theorem in the case of number fields and that analogues also exist in the space of real symplectic lattices. My question is whether there exist any analogues of Siegel's Theorem for other homogeneous spaces. I would also be grateful if someone could provide me with a reference for proofs of the Siegel Mean Value Theorem that are written in notation and style more contemporary than Siegel's original paper.

Thank you!

number-theory reference-request harmonic-analysis lattices-in-lie-groups arithmetic-dynamics

share|cite|improve this question

asked 2 days ago

Mishel SkenderiMishel Skenderi

1699

$endgroup$

share|cite|improve this question

asked 2 days ago

Mishel SkenderiMishel Skenderi

1699

share|cite|improve this question

asked 2 days ago

Mishel SkenderiMishel Skenderi

1699

asked 2 days ago

Mishel SkenderiMishel Skenderi

1699

asked 2 days ago

Mishel SkenderiMishel Skenderi

1699