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psl_2 changed to sl_2

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edited Apr 28 2011 at 19:09

Marc Palm
6,946●7●20

That's how I see it. Please correct me if I am wrong:

Induce the trivial representation from a lattice $\Gamma$, e.g. $\mathrm{PSL}_2( \mathrm{SL}_2( \mathbb{Z})$, to the group $\mathrm{PSL}_2(\mathbb{R})$.\mathrm{SL}_2(\mathbb{R})$.
Then consider the $\mathrm{PSO}_2(\mathbb{R})$-invariant \mathrm{SO}_2(\mathbb{R})$-invariant subspace.(weights are actually associated to one dimensional representations of PSO_2SO_2)

This representation is isomorphic to the representation $L^2(D,dz)$, you construct above.

How does the Laplace Beltrami operator enter the picture... It is the descent of the Casimir operator to this space, which is the generator of all invariant differential operators (= center of the universal envelopping algebra of \mathfrak{sl}_2(\mathbb{R})). So intuitevely the Casimir operator captures the $G$ structure on the Hilbert space.

Why did we became interested in such constructions? Maass discovered that the Mellin transforms of Dedekind zeta function associated to a quadratic real fields are Maass forms.

Now, since all interesting $L$ functions (ass. to Galois repr., elliptic curves,...) are conjectured to be associated to some representation to some group, it seems worthwhile: 1. To study if and how they are associated ... (e.g., Taniyama-Shimura conjecture) 2. To study their properties on either side and conclude about the other...

Mappings between groups and comparing their L-functions give also nice information about them (functoriality), e.g. they generalized Ramanujan conjecture would be implied by certain "functoriality" conjectures betweens general linear groups .

Perhaps I should conclude that with the adeles $\mathbb{A}$ the better picture is $$ \mathcal{L}^{2} ( \mathrm{PSL}_2(\mathbb{Q}) mathrm{SL}_2(\mathbb{Q}) \backslash \mathrm{PSL}_2(\mathbb{A}))^{\mathrm{K}(m)\mathrm{PSO}(2)} mathrm{SL}_2(\mathbb{A}))^{\mathrm{K}(m)\mathrm{SO}(2)} \cong \mathcal{L}^2(\Gamma(m) \backslash \mathbb{H}),$$

where $\mathrm{K}(m)$ is the product over $\mathrm{K}_p(m)$, the group of elements in $\mathrm{PSL}_2( \mathrm{SL}_2( \mathbb{Z}_p)$ which are the identy modulo $m$. This picture contains the Hecke operators more naturally...

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edited Apr 1 2011 at 23:47

David Hansen
8,389●2●18●48

That's how I see it. Please correct me if I am wrong:

Induce the trivial representation from a lattice $\Gamma$, e.g. $\mathrm{PSL}_2( \mathbb{Z})$, to the group $\mathrm{PSL}_2(\mathbb{R})$.
Then consider the $\mathrm{PSO}_2(\mathbb{R})$-invariant subspace.(weights are actually associated to one dimensional representations of PSO_2)

This representation is isomorphic to the representation $L^2(D,dz)$, you construct above.

Why did we became interested in such constructions? Maass discovered that the Mellin transforms of Dedekind zeta function associated to a quadratic real fields are Maass forms.

Perhaps I should conclude that with the adeles $\mathbb{A}$ the better picture is $$ \mathcal{L}^{2} ( \mathrm{PSL}_2(\mathbb{Q}) \backslash \mathrm{PSL}_2(\mathbb{A}))^{\mathrm{K}(m)\mathrm{PSO}(2)} \cong \mathcal{L}^2(\Gamma(m) \backslash \mathbb{H}),$$

where $\mathrm{K}(m)$ is the product over $\mathrm{K}_p(m)$, the group of elements in $\mathrm{PSL}_2( \mathbb{Z}_p)$ which are the identy modulo $m$. This picture contains the Hecke operators more naturally...

added 3 characters in body; added 16 characters in body

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edited Apr 1 2011 at 12:08

Marc Palm
6,946●7●20

That's how I see it. Please correct me if I am wrong:

Induce the trivial representation from a lattice $\Gamma$, e.g. $\mathrm{PSL}_2( \mathbb{Z})$, to the group $\mathrm{PSL}_2(\mathbb{R})$.
Then consider the $\mathrm{PSO}_2(\mathbb{R})$-invariant subspace.(weights are actually associated to one dimensional representations of PSO_2)

This representation is isomorphic to the representation $L^2(D,dz)$, you construct above.

How does the Laplace Beltrami operator enter the picture... It is the descent of the Casimir operator to this space, which is the generator of all invariant differential operators (= center of the universal envelopping algebra \mathfrak{sl}_2(\mathbb{R})). So intuitevely the Casimir operator captures the $G$ structure on the Hilbert space.

Why did we became interested in such constructions? Maass discovered that the Mellin transforms of Dedekind zeta function associated to a quadratic real fields are Maass forms.

deleted 9 characters in body

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edited Apr 1 2011 at 9:29

Marc Palm
6,946●7●20

That's how I see. Please correct me if I am wrong:

Induce the trivial representation from the a lattice $\Gamma$, e.g. $\mathrm{PSL}_2( \mathbb{Z})$, to the group $\mathrm{PSL}_2(\mathbb{R})$.
Then consider then consider the $\mathrm{PSO}_2(\mathbb{R})$-invariant subspace.(weights are actually associated to one dimensional representations of PSO_2)

This representation is isomorphic to the representation $L^2(D,dz)$, you construct above.

How does the Laplace Beltrami operator enter the picture... It is the descent of the Casimir operator to this space, which is the generator of all invariant differential operators (= center of the universal envelopping algebra \mathfrak{sl}_2(\mathbb{R})). So intuitevely the Casimir operator captures the $G$ structure on the Hilbert space.

Why did we became interested in such constructions? Maass discovered that the Mellin transforms of Dedekind zeta function associated to a quadratic real fields.

deleted 22 characters in body

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edited Apr 1 2011 at 9:05

Marc Palm
6,946●7●20

That's how I see. Please correct me if I am wrong:

Induce the trivial representation from the lattice $\Gamma$, e.g. $\mathrm{PSL}_2( \mathbb{Z})$, to the $\mathrm{PSL}_2(\mathbb{R})$.
Then consider then consider the $\mathrm{PSO}_2(\mathbb{R})$-invariant subspace.(weights are actually associated to one dimensional representations of PSO_2)

This representation is isomorphic to the representation $L^2(D,dz)$, you construct above.

How does the Laplace Beltrami operator enter the picture... It is the descent of the Casimir operator to this space, which is the generator of all invariant differential operators (= center of the universal envelopping algebra \mathfrak{sl}_2(\mathbb{R})). So intuitevely the Casimir operator captures the $G$ structure on the Hilbert space.

Why did we became interested in such constructions? Maass discovered that these Maass forms are the Mellin transform transforms of the Dedekind zeta function associated to a totally quadratic real fieldfields.

added 425 characters in body; deleted 1 characters in body; added 5 characters in body; added 47 characters in body; edited body; added 1 characters in body

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edited Apr 1 2011 at 9:00

Marc Palm
6,946●7●20

That's how I see. Please correct me if I am wrong.:

Induce the trivial representation from the lattice $\Gamma$, e.g. $\mathrm{PSL}_2( \mathbb{Z})$, to the $\mathrm{PSL}_2(\mathbb{R})$.
Then consider then consider the $\mathrm{PSO}_2(\mathbb{R})$-invariant subspace.(weights are actually associated to one dimensional representations of PSO_2)

This representation is isomorphic to the representation $L^2(D,dz)$, you construct above.

How does the Laplace Beltrami operator enter the picture... It is the descent of the Casimir operator to this space, which is the generator of all invariant differential operators (= center of the universal envelopping algebra \mathfrak{sl}_2(\mathbb{R})). So intuitevely the Casimir operator captures the $G$ structure on the Hilbert space.

Why are did we became interested in such constructionconstructions? Maass discovered that these Maass forms are the Mellin transform of the Dedekind zeta function associated to a totally real field.

[made Community Wiki]

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answered Apr 1 2011 at 8:37

Marc Palm
6,946●7●20